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Theorem dom2 6425
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 NM, 26-Oct-2003.)
Hypotheses
Ref Expression
dom2.1 (𝑥𝐴𝐶𝐵)
dom2.2 ((𝑥𝐴𝑦𝐴) → (𝐶 = 𝐷𝑥 = 𝑦))
Assertion
Ref Expression
dom2 (𝐵𝑉𝐴𝐵)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑦,𝐶   𝑥,𝐷
Allowed substitution hints:   𝐶(𝑥)   𝐷(𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem dom2
StepHypRef Expression
1 eqid 2085 . 2 𝐴 = 𝐴
2 dom2.1 . . . 4 (𝑥𝐴𝐶𝐵)
32a1i 9 . . 3 (𝐴 = 𝐴 → (𝑥𝐴𝐶𝐵))
4 dom2.2 . . . 4 ((𝑥𝐴𝑦𝐴) → (𝐶 = 𝐷𝑥 = 𝑦))
54a1i 9 . . 3 (𝐴 = 𝐴 → ((𝑥𝐴𝑦𝐴) → (𝐶 = 𝐷𝑥 = 𝑦)))
63, 5dom2d 6423 . 2 (𝐴 = 𝐴 → (𝐵𝑉𝐴𝐵))
71, 6ax-mp 7 1 (𝐵𝑉𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1287  wcel 1436   class class class wbr 3814  cdom 6389
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3922  ax-sep 3925  ax-pow 3977  ax-pr 4003  ax-un 4227
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2616  df-sbc 2829  df-csb 2922  df-un 2990  df-in 2992  df-ss 2999  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-iun 3709  df-br 3815  df-opab 3869  df-mpt 3870  df-id 4087  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-rn 4415  df-res 4416  df-ima 4417  df-iota 4937  df-fun 4974  df-fn 4975  df-f 4976  df-f1 4977  df-fo 4978  df-f1o 4979  df-fv 4980  df-dom 6392
This theorem is referenced by:  infpwfidom  6745
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