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Theorem dom3 7959
 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 473 . 2 ((𝐴𝑉𝐵𝑊) → 𝐴𝑉)
6 simpr 477 . 2 ((𝐴𝑉𝐵𝑊) → 𝐵𝑊)
72, 4, 5, 6dom3d 7957 1 ((𝐴𝑉𝐵𝑊) → 𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1480   ∈ wcel 1987   class class class wbr 4623   ≼ cdom 7913 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fv 5865  df-dom 7917 This theorem is referenced by:  canth2  8073  limenpsi  8095
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