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Theorem f1domg 8208
Description: The domain of a one-to-one function is dominated by its codomain. (Contributed by NM, 4-Sep-2004.)
Assertion
Ref Expression
f1domg (𝐵𝐶 → (𝐹:𝐴1-1𝐵𝐴𝐵))

Proof of Theorem f1domg
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 f1dmex 7362 . . . . 5 ((𝐹:𝐴1-1𝐵𝐵𝐶) → 𝐴 ∈ V)
2 f1f 6312 . . . . . 6 (𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
3 fex 6710 . . . . . 6 ((𝐹:𝐴𝐵𝐴 ∈ V) → 𝐹 ∈ V)
42, 3sylan 571 . . . . 5 ((𝐹:𝐴1-1𝐵𝐴 ∈ V) → 𝐹 ∈ V)
51, 4syldan 581 . . . 4 ((𝐹:𝐴1-1𝐵𝐵𝐶) → 𝐹 ∈ V)
65expcom 400 . . 3 (𝐵𝐶 → (𝐹:𝐴1-1𝐵𝐹 ∈ V))
7 f1eq1 6307 . . . 4 (𝑓 = 𝐹 → (𝑓:𝐴1-1𝐵𝐹:𝐴1-1𝐵))
87spcegv 3487 . . 3 (𝐹 ∈ V → (𝐹:𝐴1-1𝐵 → ∃𝑓 𝑓:𝐴1-1𝐵))
96, 8syli 39 . 2 (𝐵𝐶 → (𝐹:𝐴1-1𝐵 → ∃𝑓 𝑓:𝐴1-1𝐵))
10 brdomg 8198 . 2 (𝐵𝐶 → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵))
119, 10sylibrd 250 1 (𝐵𝐶 → (𝐹:𝐴1-1𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wex 1859  wcel 2156  Vcvv 3391   class class class wbr 4844  wf 6093  1-1wf1 6094  cdom 8186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pr 5096  ax-un 7175
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6060  df-fun 6099  df-fn 6100  df-f 6101  df-f1 6102  df-fo 6103  df-f1o 6104  df-fv 6105  df-dom 8190
This theorem is referenced by:  f1dom  8210  dom2d  8229  fseqen  9129  infpssrlem5  9410  hashf1  13454  vdwlem12  15909  2ndcdisj  21469  ovolicc2lem4  23497  basellem4  25020  usgriedgleord  26331  uspgredgleord  26336
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