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Theorem f1mpt 6396
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 4669 . . . 4 𝑥(𝑥𝐴𝐶)
31, 2nfcxfr 2748 . . 3 𝑥𝐹
4 nfcv 2750 . . 3 𝑦𝐹
53, 4dff13f 6394 . 2 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
61fmpt 6273 . . 3 (∀𝑥𝐴 𝐶𝐵𝐹:𝐴𝐵)
76anbi1i 726 . 2 ((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
8 f1mpt.2 . . . . . . 7 (𝑥 = 𝑦𝐶 = 𝐷)
98eleq1d 2671 . . . . . 6 (𝑥 = 𝑦 → (𝐶𝐵𝐷𝐵))
109cbvralv 3146 . . . . 5 (∀𝑥𝐴 𝐶𝐵 ↔ ∀𝑦𝐴 𝐷𝐵)
11 raaanv 4032 . . . . . 6 (∀𝑥𝐴𝑦𝐴 (𝐶𝐵𝐷𝐵) ↔ (∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐴 𝐷𝐵))
121fvmpt2 6184 . . . . . . . . . . . . . 14 ((𝑥𝐴𝐶𝐵) → (𝐹𝑥) = 𝐶)
138, 1fvmptg 6173 . . . . . . . . . . . . . 14 ((𝑦𝐴𝐷𝐵) → (𝐹𝑦) = 𝐷)
1412, 13eqeqan12d 2625 . . . . . . . . . . . . 13 (((𝑥𝐴𝐶𝐵) ∧ (𝑦𝐴𝐷𝐵)) → ((𝐹𝑥) = (𝐹𝑦) ↔ 𝐶 = 𝐷))
1514an4s 864 . . . . . . . . . . . 12 (((𝑥𝐴𝑦𝐴) ∧ (𝐶𝐵𝐷𝐵)) → ((𝐹𝑥) = (𝐹𝑦) ↔ 𝐶 = 𝐷))
1615imbi1d 329 . . . . . . . . . . 11 (((𝑥𝐴𝑦𝐴) ∧ (𝐶𝐵𝐷𝐵)) → (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ (𝐶 = 𝐷𝑥 = 𝑦)))
1716ex 448 . . . . . . . . . 10 ((𝑥𝐴𝑦𝐴) → ((𝐶𝐵𝐷𝐵) → (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ (𝐶 = 𝐷𝑥 = 𝑦))))
1817ralimdva 2944 . . . . . . . . 9 (𝑥𝐴 → (∀𝑦𝐴 (𝐶𝐵𝐷𝐵) → ∀𝑦𝐴 (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ (𝐶 = 𝐷𝑥 = 𝑦))))
19 ralbi 3049 . . . . . . . . 9 (∀𝑦𝐴 (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ (𝐶 = 𝐷𝑥 = 𝑦)) → (∀𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)))
2018, 19syl6 34 . . . . . . . 8 (𝑥𝐴 → (∀𝑦𝐴 (𝐶𝐵𝐷𝐵) → (∀𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦))))
2120ralimia 2933 . . . . . . 7 (∀𝑥𝐴𝑦𝐴 (𝐶𝐵𝐷𝐵) → ∀𝑥𝐴 (∀𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)))
22 ralbi 3049 . . . . . . 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 223 . . . . 5 ((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐴 𝐷𝐵) → (∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)))
2510, 24sylan2b 490 . . . 4 ((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑥𝐴 𝐶𝐵) → (∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)))
2625anidms 674 . . 3 (∀𝑥𝐴 𝐶𝐵 → (∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)))
2726pm5.32i 666 . 2 ((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)) ↔ (∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)))
285, 7, 273bitr2i 286 1 (𝐹:𝐴1-1𝐵 ↔ (∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wral 2895  cmpt 4637  wf 5785  1-1wf1 5786  cfv 5789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fv 5797
This theorem is referenced by:  ismon2  16165  isepi2  16172  usgraidx2v  25715  aciunf1lem  28637  disjf1  38147  uspgredg2v  40432  usgredg2v  40435
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