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Theorem dom3d 6800
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 6798 . . . . 5 (𝜑 → (𝑥𝐴𝐶):𝐴1-1𝐵)
4 f1f 5440 . . . . 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 5403 . . . 4 (((𝑥𝐴𝐶):𝐴𝐵𝐴𝑉𝐵𝑊) → (𝑥𝐴𝐶) ∈ V)
95, 6, 7, 8syl3anc 1249 . . 3 (𝜑 → (𝑥𝐴𝐶) ∈ V)
10 f1eq1 5435 . . . 4 (𝑧 = (𝑥𝐴𝐶) → (𝑧:𝐴1-1𝐵 ↔ (𝑥𝐴𝐶):𝐴1-1𝐵))
1110spcegv 2840 . . 3 ((𝑥𝐴𝐶) ∈ V → ((𝑥𝐴𝐶):𝐴1-1𝐵 → ∃𝑧 𝑧:𝐴1-1𝐵))
129, 3, 11sylc 62 . 2 (𝜑 → ∃𝑧 𝑧:𝐴1-1𝐵)
13 brdomg 6774 . . 3 (𝐵𝑊 → (𝐴𝐵 ↔ ∃𝑧 𝑧:𝐴1-1𝐵))
147, 13syl 14 . 2 (𝜑 → (𝐴𝐵 ↔ ∃𝑧 𝑧:𝐴1-1𝐵))
1512, 14mpbird 167 1 (𝜑𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1364  wex 1503  wcel 2160  Vcvv 2752   class class class wbr 4018  cmpt 4079  wf 5231  1-1wf1 5232  cdom 6765
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fv 5243  df-dom 6768
This theorem is referenced by:  dom3  6802  xpdom2  6857  fopwdom  6864
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