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Theorem f1mpt 5532
Description: Express injection for a mapping operation. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
f1mpt.1 𝐹 = (𝑥𝐴𝐶)
f1mpt.2 (𝑥 = 𝑦𝐶 = 𝐷)
Assertion
Ref Expression
f1mpt (𝐹:𝐴1-1𝐵 ↔ (∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑦,𝐶   𝑥,𝐷   𝑦,𝐹
Allowed substitution hints:   𝐶(𝑥)   𝐷(𝑦)   𝐹(𝑥)

Proof of Theorem f1mpt
StepHypRef Expression
1 f1mpt.1 . . . 4 𝐹 = (𝑥𝐴𝐶)
2 nfmpt1 3923 . . . 4 𝑥(𝑥𝐴𝐶)
31, 2nfcxfr 2225 . . 3 𝑥𝐹
4 nfcv 2228 . . 3 𝑦𝐹
53, 4dff13f 5531 . 2 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
61fmpt 5433 . . 3 (∀𝑥𝐴 𝐶𝐵𝐹:𝐴𝐵)
76anbi1i 446 . 2 ((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
8 f1mpt.2 . . . . . . 7 (𝑥 = 𝑦𝐶 = 𝐷)
98eleq1d 2156 . . . . . 6 (𝑥 = 𝑦 → (𝐶𝐵𝐷𝐵))
109cbvralv 2590 . . . . 5 (∀𝑥𝐴 𝐶𝐵 ↔ ∀𝑦𝐴 𝐷𝐵)
11 raaanv 3385 . . . . . 6 (∀𝑥𝐴𝑦𝐴 (𝐶𝐵𝐷𝐵) ↔ (∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐴 𝐷𝐵))
121fvmpt2 5370 . . . . . . . . . . . . . 14 ((𝑥𝐴𝐶𝐵) → (𝐹𝑥) = 𝐶)
138, 1fvmptg 5364 . . . . . . . . . . . . . 14 ((𝑦𝐴𝐷𝐵) → (𝐹𝑦) = 𝐷)
1412, 13eqeqan12d 2103 . . . . . . . . . . . . 13 (((𝑥𝐴𝐶𝐵) ∧ (𝑦𝐴𝐷𝐵)) → ((𝐹𝑥) = (𝐹𝑦) ↔ 𝐶 = 𝐷))
1514an4s 555 . . . . . . . . . . . 12 (((𝑥𝐴𝑦𝐴) ∧ (𝐶𝐵𝐷𝐵)) → ((𝐹𝑥) = (𝐹𝑦) ↔ 𝐶 = 𝐷))
1615imbi1d 229 . . . . . . . . . . 11 (((𝑥𝐴𝑦𝐴) ∧ (𝐶𝐵𝐷𝐵)) → (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ (𝐶 = 𝐷𝑥 = 𝑦)))
1716ex 113 . . . . . . . . . 10 ((𝑥𝐴𝑦𝐴) → ((𝐶𝐵𝐷𝐵) → (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ (𝐶 = 𝐷𝑥 = 𝑦))))
1817ralimdva 2441 . . . . . . . . 9 (𝑥𝐴 → (∀𝑦𝐴 (𝐶𝐵𝐷𝐵) → ∀𝑦𝐴 (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ (𝐶 = 𝐷𝑥 = 𝑦))))
19 ralbi 2501 . . . . . . . . 9 (∀𝑦𝐴 (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ (𝐶 = 𝐷𝑥 = 𝑦)) → (∀𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)))
2018, 19syl6 33 . . . . . . . 8 (𝑥𝐴 → (∀𝑦𝐴 (𝐶𝐵𝐷𝐵) → (∀𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦))))
2120ralimia 2436 . . . . . . 7 (∀𝑥𝐴𝑦𝐴 (𝐶𝐵𝐷𝐵) → ∀𝑥𝐴 (∀𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)))
22 ralbi 2501 . . . . . . 7 (∀𝑥𝐴 (∀𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)) → (∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)))
2321, 22syl 14 . . . . . 6 (∀𝑥𝐴𝑦𝐴 (𝐶𝐵𝐷𝐵) → (∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)))
2411, 23sylbir 133 . . . . 5 ((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐴 𝐷𝐵) → (∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)))
2510, 24sylan2b 281 . . . 4 ((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑥𝐴 𝐶𝐵) → (∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)))
2625anidms 389 . . 3 (∀𝑥𝐴 𝐶𝐵 → (∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)))
2726pm5.32i 442 . 2 ((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)) ↔ (∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)))
285, 7, 273bitr2i 206 1 (𝐹:𝐴1-1𝐵 ↔ (∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1289  wcel 1438  wral 2359  cmpt 3891  wf 4998  1-1wf1 4999  cfv 5002
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fv 5010
This theorem is referenced by:  1domsn  6515
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