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Theorem dom3d 8399
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 8397 . . . . 5 (𝜑 → (𝑥𝐴𝐶):𝐴1-1𝐵)
4 f1f 6443 . . . . 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 7494 . . . 4 (((𝑥𝐴𝐶):𝐴𝐵𝐴𝑉𝐵𝑊) → (𝑥𝐴𝐶) ∈ V)
95, 6, 7, 8syl3anc 1364 . . 3 (𝜑 → (𝑥𝐴𝐶) ∈ V)
10 f1eq1 6438 . . 3 (𝑧 = (𝑥𝐴𝐶) → (𝑧:𝐴1-1𝐵 ↔ (𝑥𝐴𝐶):𝐴1-1𝐵))
119, 3, 10elabd 3606 . 2 (𝜑 → ∃𝑧 𝑧:𝐴1-1𝐵)
12 brdomg 8367 . . 3 (𝐵𝑊 → (𝐴𝐵 ↔ ∃𝑧 𝑧:𝐴1-1𝐵))
137, 12syl 17 . 2 (𝜑 → (𝐴𝐵 ↔ ∃𝑧 𝑧:𝐴1-1𝐵))
1411, 13mpbird 258 1 (𝜑𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1522  wex 1761  wcel 2081  Vcvv 3437   class class class wbr 4962  cmpt 5041  wf 6221  1-1wf1 6222  cdom 8355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fv 6233  df-dom 8359
This theorem is referenced by:  dom3  8401  xpdom2  8459  fopwdom  8472
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