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Theorem dom3d 6254
Description: A mapping (first hypothesis) that is one-to-one (second hypothesis) implies its domain is dominated by its codomain. (Contributed by Mario Carneiro, 20-May-2013.)
Hypotheses
Ref Expression
dom2d.1 (𝜑 → (𝑥𝐴𝐶𝐵))
dom2d.2 (𝜑 → ((𝑥𝐴𝑦𝐴) → (𝐶 = 𝐷𝑥 = 𝑦)))
dom3d.3 (𝜑𝐴𝑉)
dom3d.4 (𝜑𝐵𝑊)
Assertion
Ref Expression
dom3d (𝜑𝐴𝐵)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑦,𝐶   𝑥,𝐷   𝜑,𝑥,𝑦
Allowed substitution hints:   𝐶(𝑥)   𝐷(𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem dom3d
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 dom2d.1 . . . . . 6 (𝜑 → (𝑥𝐴𝐶𝐵))
2 dom2d.2 . . . . . 6 (𝜑 → ((𝑥𝐴𝑦𝐴) → (𝐶 = 𝐷𝑥 = 𝑦)))
31, 2dom2lem 6252 . . . . 5 (𝜑 → (𝑥𝐴𝐶):𝐴1-1𝐵)
4 f1f 5092 . . . . 5 ((𝑥𝐴𝐶):𝐴1-1𝐵 → (𝑥𝐴𝐶):𝐴𝐵)
53, 4syl 14 . . . 4 (𝜑 → (𝑥𝐴𝐶):𝐴𝐵)
6 dom3d.3 . . . 4 (𝜑𝐴𝑉)
7 dom3d.4 . . . 4 (𝜑𝐵𝑊)
8 fex2 5059 . . . 4 (((𝑥𝐴𝐶):𝐴𝐵𝐴𝑉𝐵𝑊) → (𝑥𝐴𝐶) ∈ V)
95, 6, 7, 8syl3anc 1135 . . 3 (𝜑 → (𝑥𝐴𝐶) ∈ V)
10 f1eq1 5087 . . . 4 (𝑧 = (𝑥𝐴𝐶) → (𝑧:𝐴1-1𝐵 ↔ (𝑥𝐴𝐶):𝐴1-1𝐵))
1110spcegv 2641 . . 3 ((𝑥𝐴𝐶) ∈ V → ((𝑥𝐴𝐶):𝐴1-1𝐵 → ∃𝑧 𝑧:𝐴1-1𝐵))
129, 3, 11sylc 56 . 2 (𝜑 → ∃𝑧 𝑧:𝐴1-1𝐵)
13 brdomg 6229 . . 3 (𝐵𝑊 → (𝐴𝐵 ↔ ∃𝑧 𝑧:𝐴1-1𝐵))
147, 13syl 14 . 2 (𝜑 → (𝐴𝐵 ↔ ∃𝑧 𝑧:𝐴1-1𝐵))
1512, 14mpbird 156 1 (𝜑𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98   = wceq 1243  wex 1381  wcel 1393  Vcvv 2557   class class class wbr 3764  cmpt 3818  wf 4898  1-1wf1 4899  cdom 6220
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fv 4910  df-dom 6223
This theorem is referenced by:  dom3  6256  xpdom2  6305  fopwdom  6310
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