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Theorem dom3 8540
Description: A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. 𝐶 and 𝐷 can be read 𝐶(𝑥) and 𝐷(𝑦), as can be inferred from their distinct variable conditions. (Contributed by Mario Carneiro, 20-May-2013.)
Hypotheses
Ref Expression
dom2.1 (𝑥𝐴𝐶𝐵)
dom2.2 ((𝑥𝐴𝑦𝐴) → (𝐶 = 𝐷𝑥 = 𝑦))
Assertion
Ref Expression
dom3 ((𝐴𝑉𝐵𝑊) → 𝐴𝐵)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑦,𝐶   𝑥,𝐷   𝑥,𝑉,𝑦   𝑥,𝑊,𝑦
Allowed substitution hints:   𝐶(𝑥)   𝐷(𝑦)

Proof of Theorem dom3
StepHypRef Expression
1 dom2.1 . . 3 (𝑥𝐴𝐶𝐵)
21a1i 11 . 2 ((𝐴𝑉𝐵𝑊) → (𝑥𝐴𝐶𝐵))
3 dom2.2 . . 3 ((𝑥𝐴𝑦𝐴) → (𝐶 = 𝐷𝑥 = 𝑦))
43a1i 11 . 2 ((𝐴𝑉𝐵𝑊) → ((𝑥𝐴𝑦𝐴) → (𝐶 = 𝐷𝑥 = 𝑦)))
5 simpl 486 . 2 ((𝐴𝑉𝐵𝑊) → 𝐴𝑉)
6 simpr 488 . 2 ((𝐴𝑉𝐵𝑊) → 𝐵𝑊)
72, 4, 5, 6dom3d 8538 1 ((𝐴𝑉𝐵𝑊) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112   class class class wbr 5033  cdom 8494
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  ax-un 7445
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-pw 4502  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  df-dom 8498
This theorem is referenced by:  canth2  8658  limenpsi  8680
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