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Theorem dom3d 8538
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 8536 . . . . 5 (𝜑 → (𝑥𝐴𝐶):𝐴1-1𝐵)
4 f1f 6553 . . . . 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 7624 . . . 4 (((𝑥𝐴𝐶):𝐴𝐵𝐴𝑉𝐵𝑊) → (𝑥𝐴𝐶) ∈ V)
95, 6, 7, 8syl3anc 1368 . . 3 (𝜑 → (𝑥𝐴𝐶) ∈ V)
10 f1eq1 6548 . . 3 (𝑧 = (𝑥𝐴𝐶) → (𝑧:𝐴1-1𝐵 ↔ (𝑥𝐴𝐶):𝐴1-1𝐵))
119, 3, 10spcedv 3550 . 2 (𝜑 → ∃𝑧 𝑧:𝐴1-1𝐵)
12 brdomg 8506 . . 3 (𝐵𝑊 → (𝐴𝐵 ↔ ∃𝑧 𝑧:𝐴1-1𝐵))
137, 12syl 17 . 2 (𝜑 → (𝐴𝐵 ↔ ∃𝑧 𝑧:𝐴1-1𝐵))
1411, 13mpbird 260 1 (𝜑𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wex 1781  wcel 2112  Vcvv 3444   class class class wbr 5033  cmpt 5113  wf 6324  1-1wf1 6325  cdom 8494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fv 6336  df-dom 8498
This theorem is referenced by:  dom3  8540  xpdom2  8599  fopwdom  8612
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