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Theorem dom2lem 8346
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 3131 . . 3 (𝜑 → ∀𝑥𝐴 𝐶𝐵)
3 eqid 2778 . . . 4 (𝑥𝐴𝐶) = (𝑥𝐴𝐶)
43fmpt 6697 . . 3 (∀𝑥𝐴 𝐶𝐵 ↔ (𝑥𝐴𝐶):𝐴𝐵)
52, 4sylib 210 . 2 (𝜑 → (𝑥𝐴𝐶):𝐴𝐵)
61imp 398 . . . . . . 7 ((𝜑𝑥𝐴) → 𝐶𝐵)
73fvmpt2 6605 . . . . . . . 8 ((𝑥𝐴𝐶𝐵) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶)
87adantll 701 . . . . . . 7 (((𝜑𝑥𝐴) ∧ 𝐶𝐵) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶)
96, 8mpdan 674 . . . . . 6 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶)
109adantrr 704 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶)
11 nfv 1873 . . . . . . . 8 𝑥(𝜑𝑦𝐴)
12 nffvmpt1 6510 . . . . . . . . 9 𝑥((𝑥𝐴𝐶)‘𝑦)
1312nfeq1 2945 . . . . . . . 8 𝑥((𝑥𝐴𝐶)‘𝑦) = 𝐷
1411, 13nfim 1859 . . . . . . 7 𝑥((𝜑𝑦𝐴) → ((𝑥𝐴𝐶)‘𝑦) = 𝐷)
15 eleq1w 2848 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
1615anbi2d 619 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝜑𝑥𝐴) ↔ (𝜑𝑦𝐴)))
1716imbi1d 334 . . . . . . . 8 (𝑥 = 𝑦 → (((𝜑𝑥𝐴) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶) ↔ ((𝜑𝑦𝐴) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶)))
1815anbi1d 620 . . . . . . . . . . . 12 (𝑥 = 𝑦 → ((𝑥𝐴𝑦𝐴) ↔ (𝑦𝐴𝑦𝐴)))
19 anidm 557 . . . . . . . . . . . 12 ((𝑦𝐴𝑦𝐴) ↔ 𝑦𝐴)
2018, 19syl6bb 279 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((𝑥𝐴𝑦𝐴) ↔ 𝑦𝐴))
2120anbi2d 619 . . . . . . . . . 10 (𝑥 = 𝑦 → ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) ↔ (𝜑𝑦𝐴)))
22 fveq2 6499 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ((𝑥𝐴𝐶)‘𝑥) = ((𝑥𝐴𝐶)‘𝑦))
2322adantr 473 . . . . . . . . . . . 12 ((𝑥 = 𝑦 ∧ (𝜑 ∧ (𝑥𝐴𝑦𝐴))) → ((𝑥𝐴𝐶)‘𝑥) = ((𝑥𝐴𝐶)‘𝑦))
24 dom2d.2 . . . . . . . . . . . . . 14 (𝜑 → ((𝑥𝐴𝑦𝐴) → (𝐶 = 𝐷𝑥 = 𝑦)))
2524imp 398 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → (𝐶 = 𝐷𝑥 = 𝑦))
2625biimparc 472 . . . . . . . . . . . 12 ((𝑥 = 𝑦 ∧ (𝜑 ∧ (𝑥𝐴𝑦𝐴))) → 𝐶 = 𝐷)
2723, 26eqeq12d 2793 . . . . . . . . . . 11 ((𝑥 = 𝑦 ∧ (𝜑 ∧ (𝑥𝐴𝑦𝐴))) → (((𝑥𝐴𝐶)‘𝑥) = 𝐶 ↔ ((𝑥𝐴𝐶)‘𝑦) = 𝐷))
2827ex 405 . . . . . . . . . 10 (𝑥 = 𝑦 → ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → (((𝑥𝐴𝐶)‘𝑥) = 𝐶 ↔ ((𝑥𝐴𝐶)‘𝑦) = 𝐷)))
2921, 28sylbird 252 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝜑𝑦𝐴) → (((𝑥𝐴𝐶)‘𝑥) = 𝐶 ↔ ((𝑥𝐴𝐶)‘𝑦) = 𝐷)))
3029pm5.74d 265 . . . . . . . 8 (𝑥 = 𝑦 → (((𝜑𝑦𝐴) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶) ↔ ((𝜑𝑦𝐴) → ((𝑥𝐴𝐶)‘𝑦) = 𝐷)))
3117, 30bitrd 271 . . . . . . 7 (𝑥 = 𝑦 → (((𝜑𝑥𝐴) → ((𝑥𝐴𝐶)‘𝑥) = 𝐶) ↔ ((𝜑𝑦𝐴) → ((𝑥𝐴𝐶)‘𝑦) = 𝐷)))
3214, 31, 9chvar 2326 . . . . . 6 ((𝜑𝑦𝐴) → ((𝑥𝐴𝐶)‘𝑦) = 𝐷)
3332adantrl 703 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → ((𝑥𝐴𝐶)‘𝑦) = 𝐷)
3410, 33eqeq12d 2793 . . . 4 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → (((𝑥𝐴𝐶)‘𝑥) = ((𝑥𝐴𝐶)‘𝑦) ↔ 𝐶 = 𝐷))
3525biimpd 221 . . . 4 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → (𝐶 = 𝐷𝑥 = 𝑦))
3634, 35sylbid 232 . . 3 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → (((𝑥𝐴𝐶)‘𝑥) = ((𝑥𝐴𝐶)‘𝑦) → 𝑥 = 𝑦))
3736ralrimivva 3141 . 2 (𝜑 → ∀𝑥𝐴𝑦𝐴 (((𝑥𝐴𝐶)‘𝑥) = ((𝑥𝐴𝐶)‘𝑦) → 𝑥 = 𝑦))
38 nfmpt1 5025 . . 3 𝑥(𝑥𝐴𝐶)
39 nfcv 2932 . . 3 𝑦(𝑥𝐴𝐶)
4038, 39dff13f 6839 . 2 ((𝑥𝐴𝐶):𝐴1-1𝐵 ↔ ((𝑥𝐴𝐶):𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (((𝑥𝐴𝐶)‘𝑥) = ((𝑥𝐴𝐶)‘𝑦) → 𝑥 = 𝑦)))
415, 37, 40sylanbrc 575 1 (𝜑 → (𝑥𝐴𝐶):𝐴1-1𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387   = wceq 1507  wcel 2050  wral 3088  cmpt 5008  wf 6184  1-1wf1 6185  cfv 6188
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fv 6196
This theorem is referenced by:  dom2d  8347  dom3d  8348  ixpfi2  8617  infxpenc2lem1  9239  dfac12lem2  9364  4sqlem11  16147  odf1o1  18458  odf1o2  18459  dis2ndc  21772  hauspwpwf1  22299  itg1addlem4  24003  basellem3  25362  fsumvma  25491  dchrisum0fno1  25789
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