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Theorem f1mpt 7271
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 5256 . . . 4 𝑥(𝑥𝐴𝐶)
31, 2nfcxfr 2897 . . 3 𝑥𝐹
4 nfcv 2899 . . 3 𝑦𝐹
53, 4dff13f 7266 . 2 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
61fmpt 7120 . . 3 (∀𝑥𝐴 𝐶𝐵𝐹:𝐴𝐵)
76anbi1i 623 . 2 ((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
8 f1mpt.2 . . . . . . 7 (𝑥 = 𝑦𝐶 = 𝐷)
98eleq1d 2814 . . . . . 6 (𝑥 = 𝑦 → (𝐶𝐵𝐷𝐵))
109cbvralvw 3231 . . . . 5 (∀𝑥𝐴 𝐶𝐵 ↔ ∀𝑦𝐴 𝐷𝐵)
11 raaanv 4522 . . . . . 6 (∀𝑥𝐴𝑦𝐴 (𝐶𝐵𝐷𝐵) ↔ (∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐴 𝐷𝐵))
121fvmpt2 7016 . . . . . . . . . . . . . 14 ((𝑥𝐴𝐶𝐵) → (𝐹𝑥) = 𝐶)
138, 1fvmptg 7003 . . . . . . . . . . . . . 14 ((𝑦𝐴𝐷𝐵) → (𝐹𝑦) = 𝐷)
1412, 13eqeqan12d 2742 . . . . . . . . . . . . 13 (((𝑥𝐴𝐶𝐵) ∧ (𝑦𝐴𝐷𝐵)) → ((𝐹𝑥) = (𝐹𝑦) ↔ 𝐶 = 𝐷))
1514an4s 659 . . . . . . . . . . . 12 (((𝑥𝐴𝑦𝐴) ∧ (𝐶𝐵𝐷𝐵)) → ((𝐹𝑥) = (𝐹𝑦) ↔ 𝐶 = 𝐷))
1615imbi1d 341 . . . . . . . . . . 11 (((𝑥𝐴𝑦𝐴) ∧ (𝐶𝐵𝐷𝐵)) → (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ (𝐶 = 𝐷𝑥 = 𝑦)))
1716ex 412 . . . . . . . . . 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 3100 . . . . . . . . 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 3077 . . . . . . 7 (∀𝑥𝐴𝑦𝐴 (𝐶𝐵𝐷𝐵) → ∀𝑥𝐴 (∀𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)))
22 ralbi 3100 . . . . . . 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 593 . . . 4 ((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑥𝐴 𝐶𝐵) → (∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)))
2625anidms 566 . . 3 (∀𝑥𝐴 𝐶𝐵 → (∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)))
2726pm5.32i 574 . 2 ((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)) ↔ (∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)))
285, 7, 273bitr2i 299 1 (𝐹:𝐴1-1𝐵 ↔ (∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1534  wcel 2099  wral 3058  cmpt 5231  wf 6544  1-1wf1 6545  cfv 6548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fv 6556
This theorem is referenced by:  ismon2  17717  isepi2  17724  uspgredg2v  29050  usgredg2v  29053  aciunf1lem  32461  fnpreimac  32470  rlocf1  33000  zringfrac  33009  ply1degltdimlem  33320  ply1degltdim  33321  disjf1  44556
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