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Theorem dom3 8936
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 483 . 2 ((𝐴𝑉𝐵𝑊) → 𝐴𝑉)
6 simpr 485 . 2 ((𝐴𝑉𝐵𝑊) → 𝐵𝑊)
72, 4, 5, 6dom3d 8934 1 ((𝐴𝑉𝐵𝑊) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106   class class class wbr 5105  cdom 8881
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fv 6504  df-dom 8885
This theorem is referenced by:  canth2  9074  limenpsi  9096
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