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Theorem resf1o 29954
Description: Restriction of functions to a superset of their support creates a bijection. (Contributed by Thierry Arnoux, 12-Sep-2017.)
Hypotheses
Ref Expression
resf1o.1 𝑋 = {𝑓 ∈ (𝐵𝑚 𝐴) ∣ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶}
resf1o.2 𝐹 = (𝑓𝑋 ↦ (𝑓𝐶))
Assertion
Ref Expression
resf1o (((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) → 𝐹:𝑋1-1-onto→(𝐵𝑚 𝐶))
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓   𝐶,𝑓   𝑓,𝑉   𝑓,𝑊   𝑓,𝑋   𝑓,𝑍
Allowed substitution hint:   𝐹(𝑓)

Proof of Theorem resf1o
Dummy variables 𝑔 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 resf1o.2 . 2 𝐹 = (𝑓𝑋 ↦ (𝑓𝐶))
2 resexg 5619 . . 3 (𝑓𝑋 → (𝑓𝐶) ∈ V)
32adantl 473 . 2 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ 𝑓𝑋) → (𝑓𝐶) ∈ V)
4 simpr 477 . . . 4 (((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑔 ∈ (𝐵𝑚 𝐶)) → 𝑔 ∈ (𝐵𝑚 𝐶))
5 difexg 4969 . . . . . . 7 (𝐴𝑉 → (𝐴𝐶) ∈ V)
653ad2ant1 1163 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐶𝐴) → (𝐴𝐶) ∈ V)
7 snex 5064 . . . . . 6 {𝑍} ∈ V
8 xpexg 7158 . . . . . 6 (((𝐴𝐶) ∈ V ∧ {𝑍} ∈ V) → ((𝐴𝐶) × {𝑍}) ∈ V)
96, 7, 8sylancl 580 . . . . 5 ((𝐴𝑉𝐵𝑊𝐶𝐴) → ((𝐴𝐶) × {𝑍}) ∈ V)
109adantr 472 . . . 4 (((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑔 ∈ (𝐵𝑚 𝐶)) → ((𝐴𝐶) × {𝑍}) ∈ V)
11 unexg 7157 . . . 4 ((𝑔 ∈ (𝐵𝑚 𝐶) ∧ ((𝐴𝐶) × {𝑍}) ∈ V) → (𝑔 ∪ ((𝐴𝐶) × {𝑍})) ∈ V)
124, 10, 11syl2anc 579 . . 3 (((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑔 ∈ (𝐵𝑚 𝐶)) → (𝑔 ∪ ((𝐴𝐶) × {𝑍})) ∈ V)
1312adantlr 706 . 2 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ 𝑔 ∈ (𝐵𝑚 𝐶)) → (𝑔 ∪ ((𝐴𝐶) × {𝑍})) ∈ V)
14 resf1o.1 . . . . 5 𝑋 = {𝑓 ∈ (𝐵𝑚 𝐴) ∣ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶}
1514rabeq2i 3346 . . . 4 (𝑓𝑋 ↔ (𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶))
1615anbi1i 617 . . 3 ((𝑓𝑋𝑔 = (𝑓𝐶)) ↔ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶)))
17 simprr 789 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → 𝑔 = (𝑓𝐶))
18 simprll 797 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → 𝑓 ∈ (𝐵𝑚 𝐴))
19 elmapi 8082 . . . . . . . . 9 (𝑓 ∈ (𝐵𝑚 𝐴) → 𝑓:𝐴𝐵)
2018, 19syl 17 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → 𝑓:𝐴𝐵)
21 simp3 1168 . . . . . . . . 9 ((𝐴𝑉𝐵𝑊𝐶𝐴) → 𝐶𝐴)
2221ad2antrr 717 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → 𝐶𝐴)
2320, 22fssresd 6253 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → (𝑓𝐶):𝐶𝐵)
24 simp2 1167 . . . . . . . . 9 ((𝐴𝑉𝐵𝑊𝐶𝐴) → 𝐵𝑊)
25 simp1 1166 . . . . . . . . . 10 ((𝐴𝑉𝐵𝑊𝐶𝐴) → 𝐴𝑉)
2625, 21ssexd 4966 . . . . . . . . 9 ((𝐴𝑉𝐵𝑊𝐶𝐴) → 𝐶 ∈ V)
27 elmapg 8073 . . . . . . . . 9 ((𝐵𝑊𝐶 ∈ V) → ((𝑓𝐶) ∈ (𝐵𝑚 𝐶) ↔ (𝑓𝐶):𝐶𝐵))
2824, 26, 27syl2anc 579 . . . . . . . 8 ((𝐴𝑉𝐵𝑊𝐶𝐴) → ((𝑓𝐶) ∈ (𝐵𝑚 𝐶) ↔ (𝑓𝐶):𝐶𝐵))
2928ad2antrr 717 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → ((𝑓𝐶) ∈ (𝐵𝑚 𝐶) ↔ (𝑓𝐶):𝐶𝐵))
3023, 29mpbird 248 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → (𝑓𝐶) ∈ (𝐵𝑚 𝐶))
3117, 30eqeltrd 2844 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → 𝑔 ∈ (𝐵𝑚 𝐶))
32 undif 4209 . . . . . . . . . . 11 (𝐶𝐴 ↔ (𝐶 ∪ (𝐴𝐶)) = 𝐴)
3332biimpi 207 . . . . . . . . . 10 (𝐶𝐴 → (𝐶 ∪ (𝐴𝐶)) = 𝐴)
3433reseq2d 5565 . . . . . . . . 9 (𝐶𝐴 → (𝑓 ↾ (𝐶 ∪ (𝐴𝐶))) = (𝑓𝐴))
3522, 34syl 17 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → (𝑓 ↾ (𝐶 ∪ (𝐴𝐶))) = (𝑓𝐴))
36 ffn 6223 . . . . . . . . 9 (𝑓:𝐴𝐵𝑓 Fn 𝐴)
37 fnresdm 6178 . . . . . . . . 9 (𝑓 Fn 𝐴 → (𝑓𝐴) = 𝑓)
3820, 36, 373syl 18 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → (𝑓𝐴) = 𝑓)
3935, 38eqtr2d 2800 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → 𝑓 = (𝑓 ↾ (𝐶 ∪ (𝐴𝐶))))
40 resundi 5586 . . . . . . 7 (𝑓 ↾ (𝐶 ∪ (𝐴𝐶))) = ((𝑓𝐶) ∪ (𝑓 ↾ (𝐴𝐶)))
4139, 40syl6eq 2815 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → 𝑓 = ((𝑓𝐶) ∪ (𝑓 ↾ (𝐴𝐶))))
4217eqcomd 2771 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → (𝑓𝐶) = 𝑔)
43 simprlr 798 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶)
4425ad2antrr 717 . . . . . . . . . 10 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → 𝐴𝑉)
45 simplr 785 . . . . . . . . . 10 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → 𝑍𝐵)
46 eqid 2765 . . . . . . . . . . 11 (𝐵 ∖ {𝑍}) = (𝐵 ∖ {𝑍})
4746ffs2 29952 . . . . . . . . . 10 ((𝐴𝑉𝑍𝐵𝑓:𝐴𝐵) → (𝑓 supp 𝑍) = (𝑓 “ (𝐵 ∖ {𝑍})))
4844, 45, 20, 47syl3anc 1490 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → (𝑓 supp 𝑍) = (𝑓 “ (𝐵 ∖ {𝑍})))
49 sseqin2 3979 . . . . . . . . . . 11 (𝐶𝐴 ↔ (𝐴𝐶) = 𝐶)
5049biimpi 207 . . . . . . . . . 10 (𝐶𝐴 → (𝐴𝐶) = 𝐶)
5122, 50syl 17 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → (𝐴𝐶) = 𝐶)
5243, 48, 513sstr4d 3808 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → (𝑓 supp 𝑍) ⊆ (𝐴𝐶))
53 simpl 474 . . . . . . . . . . . 12 ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ 𝑍𝐵) → 𝑓 ∈ (𝐵𝑚 𝐴))
5453, 19, 363syl 18 . . . . . . . . . . 11 ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ 𝑍𝐵) → 𝑓 Fn 𝐴)
55 inundif 4206 . . . . . . . . . . . 12 ((𝐴𝐶) ∪ (𝐴𝐶)) = 𝐴
5655fneq2i 6164 . . . . . . . . . . 11 (𝑓 Fn ((𝐴𝐶) ∪ (𝐴𝐶)) ↔ 𝑓 Fn 𝐴)
5754, 56sylibr 225 . . . . . . . . . 10 ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ 𝑍𝐵) → 𝑓 Fn ((𝐴𝐶) ∪ (𝐴𝐶)))
58 vex 3353 . . . . . . . . . . 11 𝑓 ∈ V
5958a1i 11 . . . . . . . . . 10 ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ 𝑍𝐵) → 𝑓 ∈ V)
60 simpr 477 . . . . . . . . . 10 ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ 𝑍𝐵) → 𝑍𝐵)
61 inindif 29802 . . . . . . . . . . 11 ((𝐴𝐶) ∩ (𝐴𝐶)) = ∅
6261a1i 11 . . . . . . . . . 10 ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ 𝑍𝐵) → ((𝐴𝐶) ∩ (𝐴𝐶)) = ∅)
63 fnsuppres 7525 . . . . . . . . . 10 ((𝑓 Fn ((𝐴𝐶) ∪ (𝐴𝐶)) ∧ (𝑓 ∈ V ∧ 𝑍𝐵) ∧ ((𝐴𝐶) ∩ (𝐴𝐶)) = ∅) → ((𝑓 supp 𝑍) ⊆ (𝐴𝐶) ↔ (𝑓 ↾ (𝐴𝐶)) = ((𝐴𝐶) × {𝑍})))
6457, 59, 60, 62, 63syl121anc 1494 . . . . . . . . 9 ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ 𝑍𝐵) → ((𝑓 supp 𝑍) ⊆ (𝐴𝐶) ↔ (𝑓 ↾ (𝐴𝐶)) = ((𝐴𝐶) × {𝑍})))
6518, 45, 64syl2anc 579 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → ((𝑓 supp 𝑍) ⊆ (𝐴𝐶) ↔ (𝑓 ↾ (𝐴𝐶)) = ((𝐴𝐶) × {𝑍})))
6652, 65mpbid 223 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → (𝑓 ↾ (𝐴𝐶)) = ((𝐴𝐶) × {𝑍}))
6742, 66uneq12d 3930 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → ((𝑓𝐶) ∪ (𝑓 ↾ (𝐴𝐶))) = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))
6841, 67eqtrd 2799 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))
6931, 68jca 507 . . . 4 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶))) → (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍}))))
7024ad2antrr 717 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → 𝐵𝑊)
7125ad2antrr 717 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → 𝐴𝑉)
72 elmapi 8082 . . . . . . . . 9 (𝑔 ∈ (𝐵𝑚 𝐶) → 𝑔:𝐶𝐵)
7372ad2antrl 719 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → 𝑔:𝐶𝐵)
74 simplr 785 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → 𝑍𝐵)
75 fconst6g 6276 . . . . . . . . 9 (𝑍𝐵 → ((𝐴𝐶) × {𝑍}):(𝐴𝐶)⟶𝐵)
7674, 75syl 17 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → ((𝐴𝐶) × {𝑍}):(𝐴𝐶)⟶𝐵)
77 disjdif 4200 . . . . . . . . 9 (𝐶 ∩ (𝐴𝐶)) = ∅
7877a1i 11 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → (𝐶 ∩ (𝐴𝐶)) = ∅)
79 fun2 6249 . . . . . . . 8 (((𝑔:𝐶𝐵 ∧ ((𝐴𝐶) × {𝑍}):(𝐴𝐶)⟶𝐵) ∧ (𝐶 ∩ (𝐴𝐶)) = ∅) → (𝑔 ∪ ((𝐴𝐶) × {𝑍})):(𝐶 ∪ (𝐴𝐶))⟶𝐵)
8073, 76, 78, 79syl21anc 866 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → (𝑔 ∪ ((𝐴𝐶) × {𝑍})):(𝐶 ∪ (𝐴𝐶))⟶𝐵)
81 simprr 789 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))
8281eqcomd 2771 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → (𝑔 ∪ ((𝐴𝐶) × {𝑍})) = 𝑓)
8321ad2antrr 717 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → 𝐶𝐴)
8483, 33syl 17 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → (𝐶 ∪ (𝐴𝐶)) = 𝐴)
8582, 84feq12d 6211 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → ((𝑔 ∪ ((𝐴𝐶) × {𝑍})):(𝐶 ∪ (𝐴𝐶))⟶𝐵𝑓:𝐴𝐵))
8680, 85mpbid 223 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → 𝑓:𝐴𝐵)
87 elmapg 8073 . . . . . . 7 ((𝐵𝑊𝐴𝑉) → (𝑓 ∈ (𝐵𝑚 𝐴) ↔ 𝑓:𝐴𝐵))
8887biimpar 469 . . . . . 6 (((𝐵𝑊𝐴𝑉) ∧ 𝑓:𝐴𝐵) → 𝑓 ∈ (𝐵𝑚 𝐴))
8970, 71, 86, 88syl21anc 866 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → 𝑓 ∈ (𝐵𝑚 𝐴))
9071, 74, 86, 47syl3anc 1490 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → (𝑓 supp 𝑍) = (𝑓 “ (𝐵 ∖ {𝑍})))
9181adantr 472 . . . . . . . . 9 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))
9291fveq1d 6377 . . . . . . . 8 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → (𝑓𝑥) = ((𝑔 ∪ ((𝐴𝐶) × {𝑍}))‘𝑥))
9373adantr 472 . . . . . . . . . 10 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → 𝑔:𝐶𝐵)
9493ffnd 6224 . . . . . . . . 9 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → 𝑔 Fn 𝐶)
95 fconstg 6274 . . . . . . . . . . 11 (𝑍𝐵 → ((𝐴𝐶) × {𝑍}):(𝐴𝐶)⟶{𝑍})
9695ad3antlr 722 . . . . . . . . . 10 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → ((𝐴𝐶) × {𝑍}):(𝐴𝐶)⟶{𝑍})
9796ffnd 6224 . . . . . . . . 9 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → ((𝐴𝐶) × {𝑍}) Fn (𝐴𝐶))
9877a1i 11 . . . . . . . . 9 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → (𝐶 ∩ (𝐴𝐶)) = ∅)
99 simpr 477 . . . . . . . . 9 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → 𝑥 ∈ (𝐴𝐶))
100 fvun2 6459 . . . . . . . . 9 ((𝑔 Fn 𝐶 ∧ ((𝐴𝐶) × {𝑍}) Fn (𝐴𝐶) ∧ ((𝐶 ∩ (𝐴𝐶)) = ∅ ∧ 𝑥 ∈ (𝐴𝐶))) → ((𝑔 ∪ ((𝐴𝐶) × {𝑍}))‘𝑥) = (((𝐴𝐶) × {𝑍})‘𝑥))
10194, 97, 98, 99, 100syl112anc 1493 . . . . . . . 8 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → ((𝑔 ∪ ((𝐴𝐶) × {𝑍}))‘𝑥) = (((𝐴𝐶) × {𝑍})‘𝑥))
102 fvconst 6623 . . . . . . . . 9 ((((𝐴𝐶) × {𝑍}):(𝐴𝐶)⟶{𝑍} ∧ 𝑥 ∈ (𝐴𝐶)) → (((𝐴𝐶) × {𝑍})‘𝑥) = 𝑍)
10396, 99, 102syl2anc 579 . . . . . . . 8 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → (((𝐴𝐶) × {𝑍})‘𝑥) = 𝑍)
10492, 101, 1033eqtrd 2803 . . . . . . 7 (((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) ∧ 𝑥 ∈ (𝐴𝐶)) → (𝑓𝑥) = 𝑍)
10586, 104suppss 7528 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → (𝑓 supp 𝑍) ⊆ 𝐶)
10690, 105eqsstr3d 3800 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶)
10781reseq1d 5564 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → (𝑓𝐶) = ((𝑔 ∪ ((𝐴𝐶) × {𝑍})) ↾ 𝐶))
108 res0 5569 . . . . . . . . . 10 (((𝐴𝐶) × {𝑍}) ↾ ∅) = ∅
109 res0 5569 . . . . . . . . . 10 (𝑔 ↾ ∅) = ∅
110108, 109eqtr4i 2790 . . . . . . . . 9 (((𝐴𝐶) × {𝑍}) ↾ ∅) = (𝑔 ↾ ∅)
11177reseq2i 5562 . . . . . . . . 9 (((𝐴𝐶) × {𝑍}) ↾ (𝐶 ∩ (𝐴𝐶))) = (((𝐴𝐶) × {𝑍}) ↾ ∅)
11277reseq2i 5562 . . . . . . . . 9 (𝑔 ↾ (𝐶 ∩ (𝐴𝐶))) = (𝑔 ↾ ∅)
113110, 111, 1123eqtr4ri 2798 . . . . . . . 8 (𝑔 ↾ (𝐶 ∩ (𝐴𝐶))) = (((𝐴𝐶) × {𝑍}) ↾ (𝐶 ∩ (𝐴𝐶)))
114113a1i 11 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → (𝑔 ↾ (𝐶 ∩ (𝐴𝐶))) = (((𝐴𝐶) × {𝑍}) ↾ (𝐶 ∩ (𝐴𝐶))))
115 fresaunres1 6259 . . . . . . 7 ((𝑔:𝐶𝐵 ∧ ((𝐴𝐶) × {𝑍}):(𝐴𝐶)⟶𝐵 ∧ (𝑔 ↾ (𝐶 ∩ (𝐴𝐶))) = (((𝐴𝐶) × {𝑍}) ↾ (𝐶 ∩ (𝐴𝐶)))) → ((𝑔 ∪ ((𝐴𝐶) × {𝑍})) ↾ 𝐶) = 𝑔)
11673, 76, 114, 115syl3anc 1490 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → ((𝑔 ∪ ((𝐴𝐶) × {𝑍})) ↾ 𝐶) = 𝑔)
117107, 116eqtr2d 2800 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → 𝑔 = (𝑓𝐶))
11889, 106, 117jca31 510 . . . 4 ((((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) ∧ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))) → ((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶)))
11969, 118impbida 835 . . 3 (((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) → (((𝑓 ∈ (𝐵𝑚 𝐴) ∧ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶) ∧ 𝑔 = (𝑓𝐶)) ↔ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))))
12016, 119syl5bb 274 . 2 (((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) → ((𝑓𝑋𝑔 = (𝑓𝐶)) ↔ (𝑔 ∈ (𝐵𝑚 𝐶) ∧ 𝑓 = (𝑔 ∪ ((𝐴𝐶) × {𝑍})))))
1211, 3, 13, 120f1od 7083 1 (((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) → 𝐹:𝑋1-1-onto→(𝐵𝑚 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  {crab 3059  Vcvv 3350  cdif 3729  cun 3730  cin 3731  wss 3732  c0 4079  {csn 4334  cmpt 4888   × cxp 5275  ccnv 5276  cres 5279  cima 5280   Fn wfn 6063  wf 6064  1-1-ontowf1o 6067  cfv 6068  (class class class)co 6842   supp csupp 7497  𝑚 cmap 8060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-1st 7366  df-2nd 7367  df-supp 7498  df-map 8062
This theorem is referenced by:  eulerpartgbij  30881
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