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Theorem dom2lem 8536
Description: A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. (Contributed by NM, 24-Jul-2004.)
Hypotheses
Ref Expression
dom2d.1 (𝜑 → (𝑥𝐴𝐶𝐵))
dom2d.2 (𝜑 → ((𝑥𝐴𝑦𝐴) → (𝐶 = 𝐷𝑥 = 𝑦)))
Assertion
Ref Expression
dom2lem (𝜑 → (𝑥𝐴𝐶):𝐴1-1𝐵)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑦,𝐶   𝑥,𝐷   𝜑,𝑥,𝑦
Allowed substitution hints:   𝐶(𝑥)   𝐷(𝑦)

Proof of Theorem dom2lem
StepHypRef Expression
1 dom2d.1 . . . 4 (𝜑 → (𝑥𝐴𝐶𝐵))
21ralrimiv 3151 . . 3 (𝜑 → ∀𝑥𝐴 𝐶𝐵)
3 eqid 2801 . . . 4 (𝑥𝐴𝐶) = (𝑥𝐴𝐶)
43fmpt 6855 . . 3 (∀𝑥𝐴 𝐶𝐵 ↔ (𝑥𝐴𝐶):𝐴𝐵)
52, 4sylib 221 . 2 (𝜑 → (𝑥𝐴𝐶):𝐴𝐵)
61imp 410 . . . . . . 7 ((𝜑𝑥𝐴) → 𝐶𝐵)
73fvmpt2 6760 . . . . . . . 8 ((𝑥𝐴𝐶𝐵) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶)
87adantll 713 . . . . . . 7 (((𝜑𝑥𝐴) ∧ 𝐶𝐵) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶)
96, 8mpdan 686 . . . . . 6 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶)
109adantrr 716 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶)
11 nfv 1915 . . . . . . . 8 𝑥(𝜑𝑦𝐴)
12 nffvmpt1 6660 . . . . . . . . 9 𝑥((𝑥𝐴𝐶)‘𝑦)
1312nfeq1 2973 . . . . . . . 8 𝑥((𝑥𝐴𝐶)‘𝑦) = 𝐷
1411, 13nfim 1897 . . . . . . 7 𝑥((𝜑𝑦𝐴) → ((𝑥𝐴𝐶)‘𝑦) = 𝐷)
15 eleq1w 2875 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
1615anbi2d 631 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝜑𝑥𝐴) ↔ (𝜑𝑦𝐴)))
1716imbi1d 345 . . . . . . . 8 (𝑥 = 𝑦 → (((𝜑𝑥𝐴) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶) ↔ ((𝜑𝑦𝐴) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶)))
1815anbi1d 632 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝑥𝐴𝑦𝐴) ↔ (𝑦𝐴𝑦𝐴)))
19 anidm 568 . . . . . . . . . . . 12 ((𝑦𝐴𝑦𝐴) ↔ 𝑦𝐴)
2018, 19syl6bb 290 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((𝑥𝐴𝑦𝐴) ↔ 𝑦𝐴))
2120anbi2d 631 . . . . . . . . . 10 (𝑥 = 𝑦 → ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ↔ (𝜑𝑦𝐴)))
22 fveq2 6649 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ((𝑥𝐴𝐶)‘𝑥) = ((𝑥𝐴𝐶)‘𝑦))
2322adantr 484 . . . . . . . . . . . 12 ((𝑥 = 𝑦 ∧ (𝜑 ∧ (𝑥𝐴𝑦𝐴))) → ((𝑥𝐴𝐶)‘𝑥) = ((𝑥𝐴𝐶)‘𝑦))
24 dom2d.2 . . . . . . . . . . . . . 14 (𝜑 → ((𝑥𝐴𝑦𝐴) → (𝐶 = 𝐷𝑥 = 𝑦)))
2524imp 410 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → (𝐶 = 𝐷𝑥 = 𝑦))
2625biimparc 483 . . . . . . . . . . . 12 ((𝑥 = 𝑦 ∧ (𝜑 ∧ (𝑥𝐴𝑦𝐴))) → 𝐶 = 𝐷)
2723, 26eqeq12d 2817 . . . . . . . . . . 11 ((𝑥 = 𝑦 ∧ (𝜑 ∧ (𝑥𝐴𝑦𝐴))) → (((𝑥𝐴𝐶)‘𝑥) = 𝐶 ↔ ((𝑥𝐴𝐶)‘𝑦) = 𝐷))
2827ex 416 . . . . . . . . . 10 (𝑥 = 𝑦 → ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → (((𝑥𝐴𝐶)‘𝑥) = 𝐶 ↔ ((𝑥𝐴𝐶)‘𝑦) = 𝐷)))
2921, 28sylbird 263 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝜑𝑦𝐴) → (((𝑥𝐴𝐶)‘𝑥) = 𝐶 ↔ ((𝑥𝐴𝐶)‘𝑦) = 𝐷)))
3029pm5.74d 276 . . . . . . . 8 (𝑥 = 𝑦 → (((𝜑𝑦𝐴) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶) ↔ ((𝜑𝑦𝐴) → ((𝑥𝐴𝐶)‘𝑦) = 𝐷)))
3117, 30bitrd 282 . . . . . . 7 (𝑥 = 𝑦 → (((𝜑𝑥𝐴) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶) ↔ ((𝜑𝑦𝐴) → ((𝑥𝐴𝐶)‘𝑦) = 𝐷)))
3214, 31, 9chvarfv 2241 . . . . . 6 ((𝜑𝑦𝐴) → ((𝑥𝐴𝐶)‘𝑦) = 𝐷)
3332adantrl 715 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → ((𝑥𝐴𝐶)‘𝑦) = 𝐷)
3410, 33eqeq12d 2817 . . . 4 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → (((𝑥𝐴𝐶)‘𝑥) = ((𝑥𝐴𝐶)‘𝑦) ↔ 𝐶 = 𝐷))
3525biimpd 232 . . . 4 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → (𝐶 = 𝐷𝑥 = 𝑦))
3634, 35sylbid 243 . . 3 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → (((𝑥𝐴𝐶)‘𝑥) = ((𝑥𝐴𝐶)‘𝑦) → 𝑥 = 𝑦))
3736ralrimivva 3159 . 2 (𝜑 → ∀𝑥𝐴𝑦𝐴 (((𝑥𝐴𝐶)‘𝑥) = ((𝑥𝐴𝐶)‘𝑦) → 𝑥 = 𝑦))
38 nfmpt1 5131 . . 3 𝑥(𝑥𝐴𝐶)
39 nfcv 2958 . . 3 𝑦(𝑥𝐴𝐶)
4038, 39dff13f 6996 . 2 ((𝑥𝐴𝐶):𝐴1-1𝐵 ↔ ((𝑥𝐴𝐶):𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (((𝑥𝐴𝐶)‘𝑥) = ((𝑥𝐴𝐶)‘𝑦) → 𝑥 = 𝑦)))
415, 37, 40sylanbrc 586 1 (𝜑 → (𝑥𝐴𝐶):𝐴1-1𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  wral 3109  cmpt 5113  wf 6324  1-1wf1 6325  cfv 6328
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fv 6336
This theorem is referenced by:  dom2d  8537  dom3d  8538  ixpfi2  8810  infxpenc2lem1  9434  dfac12lem2  9559  4sqlem11  16285  odf1o1  18693  odf1o2  18694  dis2ndc  22069  hauspwpwf1  22596  itg1addlem4  24307  basellem3  25672  fsumvma  25801  dchrisum0fno1  26099
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