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Theorem f1mpt 7208
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 5213 . . . 4 𝑥(𝑥𝐴𝐶)
31, 2nfcxfr 2905 . . 3 𝑥𝐹
4 nfcv 2907 . . 3 𝑦𝐹
53, 4dff13f 7203 . 2 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
61fmpt 7058 . . 3 (∀𝑥𝐴 𝐶𝐵𝐹:𝐴𝐵)
76anbi1i 624 . 2 ((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
8 f1mpt.2 . . . . . . 7 (𝑥 = 𝑦𝐶 = 𝐷)
98eleq1d 2822 . . . . . 6 (𝑥 = 𝑦 → (𝐶𝐵𝐷𝐵))
109cbvralvw 3225 . . . . 5 (∀𝑥𝐴 𝐶𝐵 ↔ ∀𝑦𝐴 𝐷𝐵)
11 raaanv 4479 . . . . . 6 (∀𝑥𝐴𝑦𝐴 (𝐶𝐵𝐷𝐵) ↔ (∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐴 𝐷𝐵))
121fvmpt2 6959 . . . . . . . . . . . . . 14 ((𝑥𝐴𝐶𝐵) → (𝐹𝑥) = 𝐶)
138, 1fvmptg 6946 . . . . . . . . . . . . . 14 ((𝑦𝐴𝐷𝐵) → (𝐹𝑦) = 𝐷)
1412, 13eqeqan12d 2750 . . . . . . . . . . . . 13 (((𝑥𝐴𝐶𝐵) ∧ (𝑦𝐴𝐷𝐵)) → ((𝐹𝑥) = (𝐹𝑦) ↔ 𝐶 = 𝐷))
1514an4s 658 . . . . . . . . . . . 12 (((𝑥𝐴𝑦𝐴) ∧ (𝐶𝐵𝐷𝐵)) → ((𝐹𝑥) = (𝐹𝑦) ↔ 𝐶 = 𝐷))
1615imbi1d 341 . . . . . . . . . . 11 (((𝑥𝐴𝑦𝐴) ∧ (𝐶𝐵𝐷𝐵)) → (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ (𝐶 = 𝐷𝑥 = 𝑦)))
1716ex 413 . . . . . . . . . 10 ((𝑥𝐴𝑦𝐴) → ((𝐶𝐵𝐷𝐵) → (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ (𝐶 = 𝐷𝑥 = 𝑦))))
1817ralimdva 3164 . . . . . . . . 9 (𝑥𝐴 → (∀𝑦𝐴 (𝐶𝐵𝐷𝐵) → ∀𝑦𝐴 (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ (𝐶 = 𝐷𝑥 = 𝑦))))
19 ralbi 3106 . . . . . . . . 9 (∀𝑦𝐴 (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ (𝐶 = 𝐷𝑥 = 𝑦)) → (∀𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)))
2018, 19syl6 35 . . . . . . . 8 (𝑥𝐴 → (∀𝑦𝐴 (𝐶𝐵𝐷𝐵) → (∀𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦))))
2120ralimia 3083 . . . . . . 7 (∀𝑥𝐴𝑦𝐴 (𝐶𝐵𝐷𝐵) → ∀𝑥𝐴 (∀𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)))
22 ralbi 3106 . . . . . . 7 (∀𝑥𝐴 (∀𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)) → (∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)))
2321, 22syl 17 . . . . . 6 (∀𝑥𝐴𝑦𝐴 (𝐶𝐵𝐷𝐵) → (∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)))
2411, 23sylbir 234 . . . . 5 ((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐴 𝐷𝐵) → (∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)))
2510, 24sylan2b 594 . . . 4 ((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑥𝐴 𝐶𝐵) → (∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)))
2625anidms 567 . . 3 (∀𝑥𝐴 𝐶𝐵 → (∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)))
2726pm5.32i 575 . 2 ((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)) ↔ (∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)))
285, 7, 273bitr2i 298 1 (𝐹:𝐴1-1𝐵 ↔ (∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3064  cmpt 5188  wf 6492  1-1wf1 6493  cfv 6496
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fv 6504
This theorem is referenced by:  ismon2  17617  isepi2  17624  uspgredg2v  28172  usgredg2v  28175  aciunf1lem  31578  fnpreimac  31587  disjf1  43391
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