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Theorem f1mpt 5766
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 4093 . . . 4 𝑥(𝑥𝐴𝐶)
31, 2nfcxfr 2316 . . 3 𝑥𝐹
4 nfcv 2319 . . 3 𝑦𝐹
53, 4dff13f 5765 . 2 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
61fmpt 5662 . . 3 (∀𝑥𝐴 𝐶𝐵𝐹:𝐴𝐵)
76anbi1i 458 . 2 ((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
8 f1mpt.2 . . . . . . 7 (𝑥 = 𝑦𝐶 = 𝐷)
98eleq1d 2246 . . . . . 6 (𝑥 = 𝑦 → (𝐶𝐵𝐷𝐵))
109cbvralv 2703 . . . . 5 (∀𝑥𝐴 𝐶𝐵 ↔ ∀𝑦𝐴 𝐷𝐵)
11 raaanv 3530 . . . . . 6 (∀𝑥𝐴𝑦𝐴 (𝐶𝐵𝐷𝐵) ↔ (∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐴 𝐷𝐵))
121fvmpt2 5595 . . . . . . . . . . . . . 14 ((𝑥𝐴𝐶𝐵) → (𝐹𝑥) = 𝐶)
138, 1fvmptg 5588 . . . . . . . . . . . . . 14 ((𝑦𝐴𝐷𝐵) → (𝐹𝑦) = 𝐷)
1412, 13eqeqan12d 2193 . . . . . . . . . . . . 13 (((𝑥𝐴𝐶𝐵) ∧ (𝑦𝐴𝐷𝐵)) → ((𝐹𝑥) = (𝐹𝑦) ↔ 𝐶 = 𝐷))
1514an4s 588 . . . . . . . . . . . 12 (((𝑥𝐴𝑦𝐴) ∧ (𝐶𝐵𝐷𝐵)) → ((𝐹𝑥) = (𝐹𝑦) ↔ 𝐶 = 𝐷))
1615imbi1d 231 . . . . . . . . . . 11 (((𝑥𝐴𝑦𝐴) ∧ (𝐶𝐵𝐷𝐵)) → (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ (𝐶 = 𝐷𝑥 = 𝑦)))
1716ex 115 . . . . . . . . . 10 ((𝑥𝐴𝑦𝐴) → ((𝐶𝐵𝐷𝐵) → (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ (𝐶 = 𝐷𝑥 = 𝑦))))
1817ralimdva 2544 . . . . . . . . 9 (𝑥𝐴 → (∀𝑦𝐴 (𝐶𝐵𝐷𝐵) → ∀𝑦𝐴 (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ (𝐶 = 𝐷𝑥 = 𝑦))))
19 ralbi 2609 . . . . . . . . 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 2538 . . . . . . 7 (∀𝑥𝐴𝑦𝐴 (𝐶𝐵𝐷𝐵) → ∀𝑥𝐴 (∀𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)))
22 ralbi 2609 . . . . . . 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 135 . . . . 5 ((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐴 𝐷𝐵) → (∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)))
2510, 24sylan2b 287 . . . 4 ((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑥𝐴 𝐶𝐵) → (∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)))
2625anidms 397 . . 3 (∀𝑥𝐴 𝐶𝐵 → (∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)))
2726pm5.32i 454 . 2 ((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)) ↔ (∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)))
285, 7, 273bitr2i 208 1 (𝐹:𝐴1-1𝐵 ↔ (∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  wral 2455  cmpt 4061  wf 5208  1-1wf1 5209  cfv 5212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fv 5220
This theorem is referenced by:  1domsn  6813  difinfsnlem  7092
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