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Theorem dom3 6790
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 9 . 2 ((𝐴𝑉𝐵𝑊) → (𝑥𝐴𝐶𝐵))
3 dom2.2 . . 3 ((𝑥𝐴𝑦𝐴) → (𝐶 = 𝐷𝑥 = 𝑦))
43a1i 9 . 2 ((𝐴𝑉𝐵𝑊) → ((𝑥𝐴𝑦𝐴) → (𝐶 = 𝐷𝑥 = 𝑦)))
5 simpl 109 . 2 ((𝐴𝑉𝐵𝑊) → 𝐴𝑉)
6 simpr 110 . 2 ((𝐴𝑉𝐵𝑊) → 𝐵𝑊)
72, 4, 5, 6dom3d 6788 1 ((𝐴𝑉𝐵𝑊) → 𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1363  wcel 2158   class class class wbr 4015  cdom 6753
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fv 5236  df-dom 6756
This theorem is referenced by: (None)
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