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Theorem dom3d 8987
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 8985 . . . . 5 (𝜑 → (𝑥𝐴𝐶):𝐴1-1𝐵)
4 f1f 6778 . . . . 5 ((𝑥𝐴𝐶):𝐴1-1𝐵 → (𝑥𝐴𝐶):𝐴𝐵)
53, 4syl 17 . . . 4 (𝜑 → (𝑥𝐴𝐶):𝐴𝐵)
6 dom3d.3 . . . 4 (𝜑𝐴𝑉)
7 dom3d.4 . . . 4 (𝜑𝐵𝑊)
8 fex2 7918 . . . 4 (((𝑥𝐴𝐶):𝐴𝐵𝐴𝑉𝐵𝑊) → (𝑥𝐴𝐶) ∈ V)
95, 6, 7, 8syl3anc 1368 . . 3 (𝜑 → (𝑥𝐴𝐶) ∈ V)
10 f1eq1 6773 . . 3 (𝑧 = (𝑥𝐴𝐶) → (𝑧:𝐴1-1𝐵 ↔ (𝑥𝐴𝐶):𝐴1-1𝐵))
119, 3, 10spcedv 3580 . 2 (𝜑 → ∃𝑧 𝑧:𝐴1-1𝐵)
12 brdomg 8949 . . 3 (𝐵𝑊 → (𝐴𝐵 ↔ ∃𝑧 𝑧:𝐴1-1𝐵))
137, 12syl 17 . 2 (𝜑 → (𝐴𝐵 ↔ ∃𝑧 𝑧:𝐴1-1𝐵))
1411, 13mpbird 257 1 (𝜑𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wex 1773  wcel 2098  Vcvv 3466   class class class wbr 5139  cmpt 5222  wf 6530  1-1wf1 6531  cdom 8934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fv 6542  df-dom 8938
This theorem is referenced by:  dom3  8989  xpdom2  9064  fopwdom  9077
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