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Theorem hashf1lem1 13877
 Description: Lemma for hashf1 13880. (Contributed by Mario Carneiro, 17-Apr-2015.) (Proof shortened by AV, 14-Aug-2024.)
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 hashf1lem2.2 . . . 4 (𝜑𝐵 ∈ Fin)
2 f1setex 8452 . . . 4 (𝐵 ∈ Fin → {𝑓𝑓:(𝐴 ∪ {𝑧})–1-1𝐵} ∈ V)
31, 2syl 17 . . 3 (𝜑 → {𝑓𝑓:(𝐴 ∪ {𝑧})–1-1𝐵} ∈ V)
4 abanssr 4208 . . . 4 {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} ⊆ {𝑓𝑓:(𝐴 ∪ {𝑧})–1-1𝐵}
54a1i 11 . . 3 (𝜑 → {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} ⊆ {𝑓𝑓:(𝐴 ∪ {𝑧})–1-1𝐵})
63, 5ssexd 5198 . 2 (𝜑 → {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} ∈ V)
71difexd 5203 . 2 (𝜑 → (𝐵 ∖ ran 𝐹) ∈ V)
8 vex 3413 . . . 4 𝑔 ∈ V
9 reseq1 5822 . . . . . 6 (𝑓 = 𝑔 → (𝑓𝐴) = (𝑔𝐴))
109eqeq1d 2760 . . . . 5 (𝑓 = 𝑔 → ((𝑓𝐴) = 𝐹 ↔ (𝑔𝐴) = 𝐹))
11 f1eq1 6560 . . . . 5 (𝑓 = 𝑔 → (𝑓:(𝐴 ∪ {𝑧})–1-1𝐵𝑔:(𝐴 ∪ {𝑧})–1-1𝐵))
1210, 11anbi12d 633 . . . 4 (𝑓 = 𝑔 → (((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵) ↔ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)))
138, 12elab 3590 . . 3 (𝑔 ∈ {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} ↔ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵))
14 f1f 6565 . . . . . . 7 (𝑔:(𝐴 ∪ {𝑧})–1-1𝐵𝑔:(𝐴 ∪ {𝑧})⟶𝐵)
1514ad2antll 728 . . . . . 6 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → 𝑔:(𝐴 ∪ {𝑧})⟶𝐵)
16 ssun2 4080 . . . . . . 7 {𝑧} ⊆ (𝐴 ∪ {𝑧})
17 vex 3413 . . . . . . . 8 𝑧 ∈ V
1817snss 4679 . . . . . . 7 (𝑧 ∈ (𝐴 ∪ {𝑧}) ↔ {𝑧} ⊆ (𝐴 ∪ {𝑧}))
1916, 18mpbir 234 . . . . . 6 𝑧 ∈ (𝐴 ∪ {𝑧})
20 ffvelrn 6846 . . . . . 6 ((𝑔:(𝐴 ∪ {𝑧})⟶𝐵𝑧 ∈ (𝐴 ∪ {𝑧})) → (𝑔𝑧) ∈ 𝐵)
2115, 19, 20sylancl 589 . . . . 5 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → (𝑔𝑧) ∈ 𝐵)
22 hashf1lem2.3 . . . . . . 7 (𝜑 → ¬ 𝑧𝐴)
2322adantr 484 . . . . . 6 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → ¬ 𝑧𝐴)
24 df-ima 5541 . . . . . . . . 9 (𝑔𝐴) = ran (𝑔𝐴)
25 simprl 770 . . . . . . . . . 10 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → (𝑔𝐴) = 𝐹)
2625rneqd 5784 . . . . . . . . 9 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → ran (𝑔𝐴) = ran 𝐹)
2724, 26syl5eq 2805 . . . . . . . 8 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → (𝑔𝐴) = ran 𝐹)
2827eleq2d 2837 . . . . . . 7 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → ((𝑔𝑧) ∈ (𝑔𝐴) ↔ (𝑔𝑧) ∈ ran 𝐹))
29 simprr 772 . . . . . . . 8 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → 𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)
3019a1i 11 . . . . . . . 8 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → 𝑧 ∈ (𝐴 ∪ {𝑧}))
31 ssun1 4079 . . . . . . . . 9 𝐴 ⊆ (𝐴 ∪ {𝑧})
3231a1i 11 . . . . . . . 8 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → 𝐴 ⊆ (𝐴 ∪ {𝑧}))
33 f1elima 7019 . . . . . . . 8 ((𝑔:(𝐴 ∪ {𝑧})–1-1𝐵𝑧 ∈ (𝐴 ∪ {𝑧}) ∧ 𝐴 ⊆ (𝐴 ∪ {𝑧})) → ((𝑔𝑧) ∈ (𝑔𝐴) ↔ 𝑧𝐴))
3429, 30, 32, 33syl3anc 1368 . . . . . . 7 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → ((𝑔𝑧) ∈ (𝑔𝐴) ↔ 𝑧𝐴))
3528, 34bitr3d 284 . . . . . 6 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → ((𝑔𝑧) ∈ ran 𝐹𝑧𝐴))
3623, 35mtbird 328 . . . . 5 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → ¬ (𝑔𝑧) ∈ ran 𝐹)
3721, 36eldifd 3871 . . . 4 ((𝜑 ∧ ((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)) → (𝑔𝑧) ∈ (𝐵 ∖ ran 𝐹))
3837ex 416 . . 3 (𝜑 → (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) → (𝑔𝑧) ∈ (𝐵 ∖ ran 𝐹)))
3913, 38syl5bi 245 . 2 (𝜑 → (𝑔 ∈ {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} → (𝑔𝑧) ∈ (𝐵 ∖ ran 𝐹)))
40 hashf1lem1.5 . . . . . . 7 (𝜑𝐹:𝐴1-1𝐵)
41 f1f 6565 . . . . . . 7 (𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
4240, 41syl 17 . . . . . 6 (𝜑𝐹:𝐴𝐵)
4342adantr 484 . . . . 5 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → 𝐹:𝐴𝐵)
44 vex 3413 . . . . . . . 8 𝑥 ∈ V
4517, 44f1osn 6646 . . . . . . 7 {⟨𝑧, 𝑥⟩}:{𝑧}–1-1-onto→{𝑥}
46 f1of 6607 . . . . . . 7 ({⟨𝑧, 𝑥⟩}:{𝑧}–1-1-onto→{𝑥} → {⟨𝑧, 𝑥⟩}:{𝑧}⟶{𝑥})
4745, 46ax-mp 5 . . . . . 6 {⟨𝑧, 𝑥⟩}:{𝑧}⟶{𝑥}
48 eldifi 4034 . . . . . . . 8 (𝑥 ∈ (𝐵 ∖ ran 𝐹) → 𝑥𝐵)
4948adantl 485 . . . . . . 7 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → 𝑥𝐵)
5049snssd 4702 . . . . . 6 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → {𝑥} ⊆ 𝐵)
51 fss 6517 . . . . . 6 (({⟨𝑧, 𝑥⟩}:{𝑧}⟶{𝑥} ∧ {𝑥} ⊆ 𝐵) → {⟨𝑧, 𝑥⟩}:{𝑧}⟶𝐵)
5247, 50, 51sylancr 590 . . . . 5 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → {⟨𝑧, 𝑥⟩}:{𝑧}⟶𝐵)
53 res0 5832 . . . . . . 7 (𝐹 ↾ ∅) = ∅
54 res0 5832 . . . . . . 7 ({⟨𝑧, 𝑥⟩} ↾ ∅) = ∅
5553, 54eqtr4i 2784 . . . . . 6 (𝐹 ↾ ∅) = ({⟨𝑧, 𝑥⟩} ↾ ∅)
56 disjsn 4607 . . . . . . . . 9 ((𝐴 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝐴)
5722, 56sylibr 237 . . . . . . . 8 (𝜑 → (𝐴 ∩ {𝑧}) = ∅)
5857adantr 484 . . . . . . 7 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐴 ∩ {𝑧}) = ∅)
5958reseq2d 5828 . . . . . 6 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ↾ (𝐴 ∩ {𝑧})) = (𝐹 ↾ ∅))
6058reseq2d 5828 . . . . . 6 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → ({⟨𝑧, 𝑥⟩} ↾ (𝐴 ∩ {𝑧})) = ({⟨𝑧, 𝑥⟩} ↾ ∅))
6155, 59, 603eqtr4a 2819 . . . . 5 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ↾ (𝐴 ∩ {𝑧})) = ({⟨𝑧, 𝑥⟩} ↾ (𝐴 ∩ {𝑧})))
62 fresaunres1 6541 . . . . 5 ((𝐹:𝐴𝐵 ∧ {⟨𝑧, 𝑥⟩}:{𝑧}⟶𝐵 ∧ (𝐹 ↾ (𝐴 ∩ {𝑧})) = ({⟨𝑧, 𝑥⟩} ↾ (𝐴 ∩ {𝑧}))) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴) = 𝐹)
6343, 52, 61, 62syl3anc 1368 . . . 4 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴) = 𝐹)
64 f1f1orn 6618 . . . . . . . . 9 (𝐹:𝐴1-1𝐵𝐹:𝐴1-1-onto→ran 𝐹)
6540, 64syl 17 . . . . . . . 8 (𝜑𝐹:𝐴1-1-onto→ran 𝐹)
6665adantr 484 . . . . . . 7 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → 𝐹:𝐴1-1-onto→ran 𝐹)
6745a1i 11 . . . . . . 7 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → {⟨𝑧, 𝑥⟩}:{𝑧}–1-1-onto→{𝑥})
68 eldifn 4035 . . . . . . . . 9 (𝑥 ∈ (𝐵 ∖ ran 𝐹) → ¬ 𝑥 ∈ ran 𝐹)
6968adantl 485 . . . . . . . 8 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → ¬ 𝑥 ∈ ran 𝐹)
70 disjsn 4607 . . . . . . . 8 ((ran 𝐹 ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ ran 𝐹)
7169, 70sylibr 237 . . . . . . 7 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (ran 𝐹 ∩ {𝑥}) = ∅)
72 f1oun 6626 . . . . . . 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}))
7366, 67, 58, 71, 72syl22anc 837 . . . . . 6 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1-onto→(ran 𝐹 ∪ {𝑥}))
74 f1of1 6606 . . . . . 6 ((𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1-onto→(ran 𝐹 ∪ {𝑥}) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1→(ran 𝐹 ∪ {𝑥}))
7573, 74syl 17 . . . . 5 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1→(ran 𝐹 ∪ {𝑥}))
7643frnd 6510 . . . . . 6 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → ran 𝐹𝐵)
7776, 50unssd 4093 . . . . 5 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (ran 𝐹 ∪ {𝑥}) ⊆ 𝐵)
78 f1ss 6571 . . . . 5 (((𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1→(ran 𝐹 ∪ {𝑥}) ∧ (ran 𝐹 ∪ {𝑥}) ⊆ 𝐵) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1𝐵)
7975, 77, 78syl2anc 587 . . . 4 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1𝐵)
80 hashf1lem2.1 . . . . . . . 8 (𝜑𝐴 ∈ Fin)
8142, 80fexd 6987 . . . . . . 7 (𝜑𝐹 ∈ V)
8281adantr 484 . . . . . 6 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → 𝐹 ∈ V)
83 snex 5304 . . . . . 6 {⟨𝑧, 𝑥⟩} ∈ V
84 unexg 7476 . . . . . 6 ((𝐹 ∈ V ∧ {⟨𝑧, 𝑥⟩} ∈ V) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ V)
8582, 83, 84sylancl 589 . . . . 5 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ V)
86 reseq1 5822 . . . . . . . 8 (𝑓 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) → (𝑓𝐴) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴))
8786eqeq1d 2760 . . . . . . 7 (𝑓 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) → ((𝑓𝐴) = 𝐹 ↔ ((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴) = 𝐹))
88 f1eq1 6560 . . . . . . 7 (𝑓 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) → (𝑓:(𝐴 ∪ {𝑧})–1-1𝐵 ↔ (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1𝐵))
8987, 88anbi12d 633 . . . . . 6 (𝑓 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) → (((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵) ↔ (((𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↾ 𝐴) = 𝐹 ∧ (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1𝐵)))
9089elabg 3589 . . . . 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)))
9185, 90syl 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)))
9263, 79, 91mpbir2and 712 . . 3 ((𝜑𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)})
9392ex 416 . 2 (𝜑 → (𝑥 ∈ (𝐵 ∖ ran 𝐹) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ∈ {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)}))
9413anbi1i 626 . . 3 ((𝑔 ∈ {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) ↔ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)))
95 simprlr 779 . . . . . . 7 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → 𝑔:(𝐴 ∪ {𝑧})–1-1𝐵)
96 f1fn 6566 . . . . . . 7 (𝑔:(𝐴 ∪ {𝑧})–1-1𝐵𝑔 Fn (𝐴 ∪ {𝑧}))
9795, 96syl 17 . . . . . 6 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → 𝑔 Fn (𝐴 ∪ {𝑧}))
9873adantrl 715 . . . . . . 7 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1-onto→(ran 𝐹 ∪ {𝑥}))
99 f1ofn 6608 . . . . . . 7 ((𝐹 ∪ {⟨𝑧, 𝑥⟩}):(𝐴 ∪ {𝑧})–1-1-onto→(ran 𝐹 ∪ {𝑥}) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) Fn (𝐴 ∪ {𝑧}))
10098, 99syl 17 . . . . . 6 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (𝐹 ∪ {⟨𝑧, 𝑥⟩}) Fn (𝐴 ∪ {𝑧}))
101 eqfnfv 6798 . . . . . 6 ((𝑔 Fn (𝐴 ∪ {𝑧}) ∧ (𝐹 ∪ {⟨𝑧, 𝑥⟩}) Fn (𝐴 ∪ {𝑧})) → (𝑔 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↔ ∀𝑦 ∈ (𝐴 ∪ {𝑧})(𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦)))
10297, 100, 101syl2anc 587 . . . . 5 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (𝑔 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↔ ∀𝑦 ∈ (𝐴 ∪ {𝑧})(𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦)))
103 fvres 6682 . . . . . . . . . . 11 (𝑦𝐴 → ((𝑔𝐴)‘𝑦) = (𝑔𝑦))
104103eqcomd 2764 . . . . . . . . . 10 (𝑦𝐴 → (𝑔𝑦) = ((𝑔𝐴)‘𝑦))
105 simprll 778 . . . . . . . . . . 11 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (𝑔𝐴) = 𝐹)
106105fveq1d 6665 . . . . . . . . . 10 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → ((𝑔𝐴)‘𝑦) = (𝐹𝑦))
107104, 106sylan9eqr 2815 . . . . . . . . 9 (((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦𝐴) → (𝑔𝑦) = (𝐹𝑦))
10840ad2antrr 725 . . . . . . . . . . 11 (((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦𝐴) → 𝐹:𝐴1-1𝐵)
109 f1fn 6566 . . . . . . . . . . 11 (𝐹:𝐴1-1𝐵𝐹 Fn 𝐴)
110108, 109syl 17 . . . . . . . . . 10 (((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦𝐴) → 𝐹 Fn 𝐴)
11117, 44fnsn 6398 . . . . . . . . . . 11 {⟨𝑧, 𝑥⟩} Fn {𝑧}
112111a1i 11 . . . . . . . . . 10 (((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦𝐴) → {⟨𝑧, 𝑥⟩} Fn {𝑧})
11357ad2antrr 725 . . . . . . . . . 10 (((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦𝐴) → (𝐴 ∩ {𝑧}) = ∅)
114 simpr 488 . . . . . . . . . 10 (((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦𝐴) → 𝑦𝐴)
115110, 112, 113, 114fvun1d 6750 . . . . . . . . 9 (((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦𝐴) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) = (𝐹𝑦))
116107, 115eqtr4d 2796 . . . . . . . 8 (((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) ∧ 𝑦𝐴) → (𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦))
117116ralrimiva 3113 . . . . . . 7 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → ∀𝑦𝐴 (𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦))
118117biantrurd 536 . . . . . 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))))
119 ralunb 4098 . . . . . 6 (∀𝑦 ∈ (𝐴 ∪ {𝑧})(𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ↔ (∀𝑦𝐴 (𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ∧ ∀𝑦 ∈ {𝑧} (𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦)))
120118, 119bitr4di 292 . . . . 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)))
12142fdmd 6513 . . . . . . . . . . 11 (𝜑 → dom 𝐹 = 𝐴)
122121eleq2d 2837 . . . . . . . . . 10 (𝜑 → (𝑧 ∈ dom 𝐹𝑧𝐴))
12322, 122mtbird 328 . . . . . . . . 9 (𝜑 → ¬ 𝑧 ∈ dom 𝐹)
124123adantr 484 . . . . . . . 8 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → ¬ 𝑧 ∈ dom 𝐹)
125 fsnunfv 6946 . . . . . . . 8 ((𝑧 ∈ V ∧ 𝑥 ∈ V ∧ ¬ 𝑧 ∈ dom 𝐹) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧) = 𝑥)
12617, 44, 124, 125mp3an12i 1462 . . . . . . 7 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧) = 𝑥)
127126eqeq2d 2769 . . . . . 6 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → ((𝑔𝑧) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧) ↔ (𝑔𝑧) = 𝑥))
128 fveq2 6663 . . . . . . . 8 (𝑦 = 𝑧 → (𝑔𝑦) = (𝑔𝑧))
129 fveq2 6663 . . . . . . . 8 (𝑦 = 𝑧 → ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧))
130128, 129eqeq12d 2774 . . . . . . 7 (𝑦 = 𝑧 → ((𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ↔ (𝑔𝑧) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧)))
13117, 130ralsn 4579 . . . . . 6 (∀𝑦 ∈ {𝑧} (𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ↔ (𝑔𝑧) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑧))
132 eqcom 2765 . . . . . 6 (𝑥 = (𝑔𝑧) ↔ (𝑔𝑧) = 𝑥)
133127, 131, 1323bitr4g 317 . . . . 5 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (∀𝑦 ∈ {𝑧} (𝑔𝑦) = ((𝐹 ∪ {⟨𝑧, 𝑥⟩})‘𝑦) ↔ 𝑥 = (𝑔𝑧)))
134102, 120, 1333bitr2d 310 . . . 4 ((𝜑 ∧ (((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹))) → (𝑔 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↔ 𝑥 = (𝑔𝑧)))
135134ex 416 . . 3 (𝜑 → ((((𝑔𝐴) = 𝐹𝑔:(𝐴 ∪ {𝑧})–1-1𝐵) ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝑔 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↔ 𝑥 = (𝑔𝑧))))
13694, 135syl5bi 245 . 2 (𝜑 → ((𝑔 ∈ {𝑓 ∣ ((𝑓𝐴) = 𝐹𝑓:(𝐴 ∪ {𝑧})–1-1𝐵)} ∧ 𝑥 ∈ (𝐵 ∖ ran 𝐹)) → (𝑔 = (𝐹 ∪ {⟨𝑧, 𝑥⟩}) ↔ 𝑥 = (𝑔𝑧))))
1376, 7, 39, 93, 136en3d 8577 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 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2735  ∀wral 3070  Vcvv 3409   ∖ cdif 3857   ∪ cun 3858   ∩ cin 3859   ⊆ wss 3860  ∅c0 4227  {csn 4525  ⟨cop 4531   class class class wbr 5036  dom cdm 5528  ran crn 5529   ↾ cres 5530   “ cima 5531   Fn wfn 6335  ⟶wf 6336  –1-1→wf1 6337  –1-1-onto→wf1o 6339  ‘cfv 6340  (class class class)co 7156   ≈ cen 8537  Fincfn 8540  1c1 10589   + caddc 10591   ≤ cle 10727  ♯chash 13753 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1st 7699  df-2nd 7700  df-map 8424  df-en 8541 This theorem is referenced by:  hashf1lem2  13879
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