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Theorem hashf1lem1 13808
Description: Lemma for hashf1 13810. (Contributed by Mario Carneiro, 17-Apr-2015.)
Hypotheses
Ref Expression
hashf1lem2.1 (𝜑𝐴 ∈ Fin)
hashf1lem2.2 (𝜑𝐵 ∈ Fin)
hashf1lem2.3 (𝜑 → ¬ 𝑧𝐴)
hashf1lem2.4 (𝜑 → ((♯‘𝐴) + 1) ≤ (♯‘𝐵))
hashf1lem1.5 (𝜑𝐹:𝐴1-1𝐵)
Assertion
Ref Expression
hashf1lem1 (𝜑 → {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} ≈ (𝐵 ∖ ran 𝐹))
Distinct variable groups:   𝑧,𝑓   𝐴,𝑓   𝐵,𝑓   𝜑,𝑓   𝑓,𝐹
Allowed substitution hints:   𝜑(𝑧)   𝐴(𝑧)   𝐵(𝑧)   𝐹(𝑧)

Proof of Theorem hashf1lem1
Dummy variables 𝑔 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1f 6574 . . . . . 6 (𝑓:(𝐴 ∪ {𝑧})–1-1𝐵𝑓:(𝐴 ∪ {𝑧})⟶𝐵)
21adantl 482 . . . . 5 (((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵) → 𝑓:(𝐴 ∪ {𝑧})⟶𝐵)
3 hashf1lem2.2 . . . . . 6 (𝜑𝐵 ∈ Fin)
4 hashf1lem2.1 . . . . . . 7 (𝜑𝐴 ∈ Fin)
5 snfi 8588 . . . . . . 7 {𝑧} ∈ Fin
6 unfi 8779 . . . . . . 7 ((𝐴 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝐴 ∪ {𝑧}) ∈ Fin)
74, 5, 6sylancl 586 . . . . . 6 (𝜑 → (𝐴 ∪ {𝑧}) ∈ Fin)
83, 7elmapd 8415 . . . . 5 (𝜑 → (𝑓 ∈ (𝐵m (𝐴 ∪ {𝑧})) ↔ 𝑓:(𝐴 ∪ {𝑧})⟶𝐵))
92, 8syl5ibr 247 . . . 4 (𝜑 → (((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵) → 𝑓 ∈ (𝐵m (𝐴 ∪ {𝑧}))))
109abssdv 4049 . . 3 (𝜑 → {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} ⊆ (𝐵m (𝐴 ∪ {𝑧})))
11 ovex 7183 . . 3 (𝐵m (𝐴 ∪ {𝑧})) ∈ V
12 ssexg 5224 . . 3 (({𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} ⊆ (𝐵m (𝐴 ∪ {𝑧})) ∧ (𝐵m (𝐴 ∪ {𝑧})) ∈ V) → {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} ∈ V)
1310, 11, 12sylancl 586 . 2 (𝜑 → {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} ∈ V)
14 difexg 5228 . . 3 (𝐵 ∈ Fin → (𝐵 ∖ ran 𝐹) ∈ V)
153, 14syl 17 . 2 (𝜑 → (𝐵 ∖ ran 𝐹) ∈ V)
16 vex 3503 . . . 4 𝑔 ∈ V
17 reseq1 5846 . . . . . 6 (𝑓 = 𝑔 → (𝑓𝐴) = (𝑔𝐴))
1817eqeq1d 2828 . . . . 5 (𝑓 = 𝑔 → ((𝑓𝐴) = 𝐹 ↔ (𝑔𝐴) = 𝐹))
19 f1eq1 6569 . . . . 5 (𝑓 = 𝑔 → (𝑓:(𝐴 ∪ {𝑧})–1-1𝐵𝑔:(𝐴 ∪ {𝑧})–1-1𝐵))
2018, 19anbi12d 630 . . . 4 (𝑓 = 𝑔 → (((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵) ↔ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)))
2116, 20elab 3671 . . 3 (𝑔 ∈ {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} ↔ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵))
22 f1f 6574 . . . . . . 7 (𝑔:(𝐴 ∪ {𝑧})–1-1𝐵𝑔:(𝐴 ∪ {𝑧})⟶𝐵)
2322ad2antll 725 . . . . . 6 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → 𝑔:(𝐴 ∪ {𝑧})⟶𝐵)
24 ssun2 4153 . . . . . . 7 {𝑧} ⊆ (𝐴 ∪ {𝑧})
25 vex 3503 . . . . . . . 8 𝑧 ∈ V
2625snss 4717 . . . . . . 7 (𝑧 ∈ (𝐴 ∪ {𝑧}) ↔ {𝑧} ⊆ (𝐴 ∪ {𝑧}))
2724, 26mpbir 232 . . . . . 6 𝑧 ∈ (𝐴 ∪ {𝑧})
28 ffvelrn 6847 . . . . . 6 ((𝑔:(𝐴 ∪ {𝑧})⟶𝐵𝑧 ∈ (𝐴 ∪ {𝑧})) → (𝑔𝑧) ∈ 𝐵)
2923, 27, 28sylancl 586 . . . . 5 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → (𝑔𝑧) ∈ 𝐵)
30 hashf1lem2.3 . . . . . . 7 (𝜑 → ¬ 𝑧𝐴)
3130adantr 481 . . . . . 6 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → ¬ 𝑧𝐴)
32 df-ima 5567 . . . . . . . . 9 (𝑔𝐴) = ran (𝑔𝐴)
33 simprl 767 . . . . . . . . . 10 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → (𝑔𝐴) = 𝐹)
3433rneqd 5807 . . . . . . . . 9 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → ran (𝑔𝐴) = ran 𝐹)
3532, 34syl5eq 2873 . . . . . . . 8 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → (𝑔𝐴) = ran 𝐹)
3635eleq2d 2903 . . . . . . 7 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → ((𝑔𝑧) ∈ (𝑔𝐴) ↔ (𝑔𝑧) ∈ ran 𝐹))
37 simprr 769 . . . . . . . 8 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → 𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)
3827a1i 11 . . . . . . . 8 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → 𝑧 ∈ (𝐴 ∪ {𝑧}))
39 ssun1 4152 . . . . . . . . 9 𝐴 ⊆ (𝐴 ∪ {𝑧})
4039a1i 11 . . . . . . . 8 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → 𝐴 ⊆ (𝐴 ∪ {𝑧}))
41 f1elima 7017 . . . . . . . 8 ((𝑔:(𝐴 ∪ {𝑧})–1-1𝐵𝑧 ∈ (𝐴 ∪ {𝑧}) ∧ 𝐴 ⊆ (𝐴 ∪ {𝑧})) → ((𝑔𝑧) ∈ (𝑔𝐴) ↔ 𝑧𝐴))
4237, 38, 40, 41syl3anc 1365 . . . . . . 7 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → ((𝑔𝑧) ∈ (𝑔𝐴) ↔ 𝑧𝐴))
4336, 42bitr3d 282 . . . . . 6 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → ((𝑔𝑧) ∈ ran 𝐹𝑧𝐴))
4431, 43mtbird 326 . . . . 5 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → ¬ (𝑔𝑧) ∈ ran 𝐹)
4529, 44eldifd 3951 . . . 4 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → (𝑔𝑧) ∈ (𝐵 ∖ ran 𝐹))
4645ex 413 . . 3 (𝜑 → (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) → (𝑔𝑧) ∈ (𝐵 ∖ ran 𝐹)))
4721, 46syl5bi 243 . 2 (𝜑 → (𝑔 ∈ {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} → (𝑔𝑧) ∈ (𝐵 ∖ ran 𝐹)))
48 hashf1lem1.5 . . . . . . 7 (𝜑𝐹:𝐴1-1𝐵)
49 f1f 6574 . . . . . . 7 (𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
5048, 49syl 17 . . . . . 6 (𝜑𝐹:𝐴𝐵)
5150adantr 481 . . . . 5 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → 𝐹:𝐴𝐵)
52 vex 3503 . . . . . . . 8 𝑥 ∈ V
5325, 52f1osn 6653 . . . . . . 7 {⟨𝑧, 𝑥⟩}:{𝑧}–1-1-onto→{𝑥}
54 f1of 6614 . . . . . . 7 ({⟨𝑧, 𝑥⟩}:{𝑧}–1-1-onto→{𝑥} → {⟨𝑧, 𝑥⟩}:{𝑧}⟶{𝑥})
5553, 54ax-mp 5 . . . . . 6 {⟨𝑧, 𝑥⟩}:{𝑧}⟶{𝑥}
56 eldifi 4107 . . . . . . . 8 (𝑥 ∈ (𝐵 ∖ ran 𝐹) → 𝑥𝐵)
5756adantl 482 . . . . . . 7 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → 𝑥𝐵)
5857snssd 4741 . . . . . 6 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → {𝑥} ⊆ 𝐵)
59 fss 6526 . . . . . 6 (({⟨𝑧, 𝑥⟩}:{𝑧}⟶{𝑥} ∧ {𝑥} ⊆ 𝐵) → {⟨𝑧, 𝑥⟩}:{𝑧}⟶𝐵)
6055, 58, 59sylancr 587 . . . . 5 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → {⟨𝑧, 𝑥⟩}:{𝑧}⟶𝐵)
61 res0 5856 . . . . . . 7 (𝐹 ↾ ∅) = ∅
62 res0 5856 . . . . . . 7 ({⟨𝑧, 𝑥⟩} ↾ ∅) = ∅
6361, 62eqtr4i 2852 . . . . . 6 (𝐹 ↾ ∅) = ({⟨𝑧, 𝑥⟩} ↾ ∅)
64 disjsn 4646 . . . . . . . . 9 ((𝐴 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝐴)
6530, 64sylibr 235 . . . . . . . 8 (𝜑 → (𝐴 ∩ {𝑧}) = ∅)
6665adantr 481 . . . . . . 7 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐴 ∩ {𝑧}) = ∅)
6766reseq2d 5852 . . . . . 6 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ↾ (𝐴 ∩ {𝑧})) = (𝐹 ↾ ∅))
6866reseq2d 5852 . . . . . 6 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → ({⟨𝑧, 𝑥⟩} ↾ (𝐴 ∩ {𝑧})) = ({⟨𝑧, 𝑥⟩} ↾ ∅))
6963, 67, 683eqtr4a 2887 . . . . 5 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ↾ (𝐴 ∩ {𝑧})) = ({⟨𝑧, 𝑥⟩} ↾ (𝐴 ∩ {𝑧})))
70 fresaunres1 6550 . . . . 5 ((𝐹:𝐴𝐵 ∧ {⟨𝑧, 𝑥⟩}:{𝑧}⟶𝐵 ∧ (𝐹 ↾ (𝐴 ∩ {𝑧})) = ({⟨𝑧, 𝑥⟩} ↾ (𝐴 ∩ {𝑧}))) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴) = 𝐹)
7151, 60, 69, 70syl3anc 1365 . . . 4 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴) = 𝐹)
72 f1f1orn 6625 . . . . . . . . 9 (𝐹:𝐴1-1𝐵𝐹:𝐴1-1-onto→ran 𝐹)
7348, 72syl 17 . . . . . . . 8 (𝜑𝐹:𝐴1-1-onto→ran 𝐹)
7473adantr 481 . . . . . . 7 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → 𝐹:𝐴1-1-onto→ran 𝐹)
7553a1i 11 . . . . . . 7 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → {⟨𝑧, 𝑥⟩}:{𝑧}–1-1-onto→{𝑥})
76 eldifn 4108 . . . . . . . . 9 (𝑥 ∈ (𝐵 ∖ ran 𝐹) → ¬ 𝑥 ∈ ran 𝐹)
7776adantl 482 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → ¬ 𝑥 ∈ ran 𝐹)
78 disjsn 4646 . . . . . . . 8 ((ran 𝐹 ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ ran 𝐹)
7977, 78sylibr 235 . . . . . . 7 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (ran 𝐹 ∩ {𝑥}) = ∅)
80 f1oun 6633 . . . . . . 7 (((𝐹:𝐴1-1-onto→ran 𝐹 ∧ {⟨𝑧, 𝑥⟩}:{𝑧}–1-1-onto→{𝑥}) ∧ ((𝐴 ∩ {𝑧}) = ∅ ∧ (ran 𝐹 ∩ {𝑥}) = ∅)) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1-onto→(ran 𝐹 ∪ {𝑥}))
8174, 75, 66, 79, 80syl22anc 836 . . . . . 6 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1-onto→(ran 𝐹 ∪ {𝑥}))
82 f1of1 6613 . . . . . 6 ((𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1-onto→(ran 𝐹 ∪ {𝑥}) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1→(ran 𝐹 ∪ {𝑥}))
8381, 82syl 17 . . . . 5 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1→(ran 𝐹 ∪ {𝑥}))
8451frnd 6520 . . . . . 6 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → ran 𝐹𝐵)
8584, 58unssd 4166 . . . . 5 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (ran 𝐹 ∪ {𝑥}) ⊆ 𝐵)
86 f1ss 6579 . . . . 5 (((𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1→(ran 𝐹 ∪ {𝑥}) ∧ (ran 𝐹 ∪ {𝑥}) ⊆ 𝐵) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1𝐵)
8783, 85, 86syl2anc 584 . . . 4 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1𝐵)
88 fex 6986 . . . . . . . 8 ((𝐹:𝐴𝐵𝐴 ∈ Fin) → 𝐹 ∈ V)
8950, 4, 88syl2anc 584 . . . . . . 7 (𝜑𝐹 ∈ V)
9089adantr 481 . . . . . 6 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → 𝐹 ∈ V)
91 snex 5328 . . . . . 6 {⟨𝑧, 𝑥⟩} ∈ V
92 unexg 7465 . . . . . 6 ((𝐹 ∈ V ∧ {⟨𝑧, 𝑥⟩} ∈ V) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ V)
9390, 91, 92sylancl 586 . . . . 5 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ V)
94 reseq1 5846 . . . . . . . 8 (𝑓 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) → (𝑓𝐴) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴))
9594eqeq1d 2828 . . . . . . 7 (𝑓 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) → ((𝑓𝐴) = 𝐹 ↔ ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴) = 𝐹))
96 f1eq1 6569 . . . . . . 7 (𝑓 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) → (𝑓:(𝐴 ∪ {𝑧})–1-1𝐵 ↔ (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1𝐵))
9795, 96anbi12d 630 . . . . . 6 (𝑓 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) → (((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵) ↔ (((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴) = 𝐹 ∧ (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1𝐵)))
9897elabg 3670 . . . . 5 ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ V → ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} ↔ (((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴) = 𝐹 ∧ (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1𝐵)))
9993, 98syl 17 . . . 4 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} ↔ (((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴) = 𝐹 ∧ (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1𝐵)))
10071, 87, 99mpbir2and 709 . . 3 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)})
101100ex 413 . 2 (𝜑 → (𝑥 ∈ (𝐵 ∖ ran 𝐹) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)}))
10221anbi1i 623 . . 3 ((𝑔 ∈ {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) ↔ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)))
103 simprlr 776 . . . . . . 7 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → 𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)
104 f1fn 6575 . . . . . . 7 (𝑔:(𝐴 ∪ {𝑧})–1-1𝐵𝑔 Fn (𝐴 ∪ {𝑧}))
105103, 104syl 17 . . . . . 6 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → 𝑔 Fn (𝐴 ∪ {𝑧}))
10681adantrl 712 . . . . . . 7 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1-onto→(ran 𝐹 ∪ {𝑥}))
107 f1ofn 6615 . . . . . . 7 ((𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1-onto→(ran 𝐹 ∪ {𝑥}) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) Fn (𝐴 ∪ {𝑧}))
108106, 107syl 17 . . . . . 6 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) Fn (𝐴 ∪ {𝑧}))
109 eqfnfv 6800 . . . . . 6 ((𝑔 Fn (𝐴 ∪ {𝑧}) ∧ (𝐹 ∪ {⟨𝑧, 𝑥⟩}) Fn (𝐴 ∪ {𝑧})) → (𝑔 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↔ ∀𝑦 ∈ (𝐴 ∪ {𝑧})(𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦)))
110105, 108, 109syl2anc 584 . . . . 5 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (𝑔 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↔ ∀𝑦 ∈ (𝐴 ∪ {𝑧})(𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦)))
111 fvres 6688 . . . . . . . . . . 11 (𝑦𝐴 → ((𝑔𝐴)‘𝑦) = (𝑔𝑦))
112111eqcomd 2832 . . . . . . . . . 10 (𝑦𝐴 → (𝑔𝑦) = ((𝑔𝐴)‘𝑦))
113 simprll 775 . . . . . . . . . . 11 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (𝑔𝐴) = 𝐹)
114113fveq1d 6671 . . . . . . . . . 10 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → ((𝑔𝐴)‘𝑦) = (𝐹𝑦))
115112, 114sylan9eqr 2883 . . . . . . . . 9 (((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦𝐴) → (𝑔𝑦) = (𝐹𝑦))
11648ad2antrr 722 . . . . . . . . . . 11 (((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦𝐴) → 𝐹:𝐴1-1𝐵)
117 f1fn 6575 . . . . . . . . . . 11 (𝐹:𝐴1-1𝐵𝐹 Fn 𝐴)
118116, 117syl 17 . . . . . . . . . 10 (((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦𝐴) → 𝐹 Fn 𝐴)
11925, 52fnsn 6411 . . . . . . . . . . 11 {⟨𝑧, 𝑥⟩} Fn {𝑧}
120119a1i 11 . . . . . . . . . 10 (((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦𝐴) → {⟨𝑧, 𝑥⟩} Fn {𝑧})
12165ad2antrr 722 . . . . . . . . . 10 (((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦𝐴) → (𝐴 ∩ {𝑧}) = ∅)
122 simpr 485 . . . . . . . . . 10 (((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦𝐴) → 𝑦𝐴)
123 fvun1 6753 . . . . . . . . . 10 ((𝐹 Fn 𝐴 ∧ {⟨𝑧, 𝑥⟩} Fn {𝑧} ∧ ((𝐴 ∩ {𝑧}) = ∅ ∧ 𝑦𝐴)) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) = (𝐹𝑦))
124118, 120, 121, 122, 123syl112anc 1368 . . . . . . . . 9 (((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦𝐴) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) = (𝐹𝑦))
125115, 124eqtr4d 2864 . . . . . . . 8 (((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦𝐴) → (𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦))
126125ralrimiva 3187 . . . . . . 7 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → ∀𝑦𝐴 (𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦))
127126biantrurd 533 . . . . . 6 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (∀𝑦 ∈ {𝑧} (𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ↔ (∀𝑦𝐴 (𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ∧ ∀𝑦 ∈ {𝑧} (𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦))))
128 ralunb 4171 . . . . . 6 (∀𝑦 ∈ (𝐴 ∪ {𝑧})(𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ↔ (∀𝑦𝐴 (𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ∧ ∀𝑦 ∈ {𝑧} (𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦)))
129127, 128syl6bbr 290 . . . . 5 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (∀𝑦 ∈ {𝑧} (𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ↔ ∀𝑦 ∈ (𝐴 ∪ {𝑧})(𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦)))
13050fdmd 6522 . . . . . . . . . . 11 (𝜑 → dom 𝐹 = 𝐴)
131130eleq2d 2903 . . . . . . . . . 10 (𝜑 → (𝑧 ∈ dom 𝐹𝑧𝐴))
13230, 131mtbird 326 . . . . . . . . 9 (𝜑 → ¬ 𝑧 ∈ dom 𝐹)
133132adantr 481 . . . . . . . 8 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → ¬ 𝑧 ∈ dom 𝐹)
134 fsnunfv 6947 . . . . . . . 8 ((𝑧 ∈ V ∧ 𝑥 ∈ V ∧ ¬ 𝑧 ∈ dom 𝐹) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧) = 𝑥)
13525, 52, 133, 134mp3an12i 1458 . . . . . . 7 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧) = 𝑥)
136135eqeq2d 2837 . . . . . 6 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → ((𝑔𝑧) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧) ↔ (𝑔𝑧) = 𝑥))
137 fveq2 6669 . . . . . . . 8 (𝑦 = 𝑧 → (𝑔𝑦) = (𝑔𝑧))
138 fveq2 6669 . . . . . . . 8 (𝑦 = 𝑧 → ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧))
139137, 138eqeq12d 2842 . . . . . . 7 (𝑦 = 𝑧 → ((𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ↔ (𝑔𝑧) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧)))
14025, 139ralsn 4618 . . . . . 6 (∀𝑦 ∈ {𝑧} (𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ↔ (𝑔𝑧) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧))
141 eqcom 2833 . . . . . 6 (𝑥 = (𝑔𝑧) ↔ (𝑔𝑧) = 𝑥)
142136, 140, 1413bitr4g 315 . . . . 5 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (∀𝑦 ∈ {𝑧} (𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ↔ 𝑥 = (𝑔𝑧)))
143110, 129, 1423bitr2d 308 . . . 4 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (𝑔 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↔ 𝑥 = (𝑔𝑧)))
144143ex 413 . . 3 (𝜑 → ((((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝑔 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↔ 𝑥 = (𝑔𝑧))))
145102, 144syl5bi 243 . 2 (𝜑 → ((𝑔 ∈ {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝑔 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↔ 𝑥 = (𝑔𝑧))))
14613, 15, 47, 101, 145en3d 8540 1 (𝜑 → {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} ≈ (𝐵 ∖ ran 𝐹))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  {cab 2804  wral 3143  Vcvv 3500  cdif 3937  cun 3938  cin 3939  wss 3940  c0 4295  {csn 4564  cop 4570   class class class wbr 5063  dom cdm 5554  ran crn 5555  cres 5556  cima 5557   Fn wfn 6349  wf 6350  1-1wf1 6351  1-1-ontowf1o 6353  cfv 6354  (class class class)co 7150  m cmap 8401  cen 8500  Fincfn 8503  1c1 10532   + caddc 10534  cle 10670  chash 13685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7574  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8284  df-map 8403  df-en 8504  df-fin 8507
This theorem is referenced by:  hashf1lem2  13809
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