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Theorem f1imaen2g 6564
Description: A one-to-one function's image under a subset of its domain is equinumerous to the subset. (This version of f1imaen 6565 does not need ax-setind 4366.) (Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario Carneiro, 25-Jun-2015.)
Assertion
Ref Expression
f1imaen2g (((𝐹:𝐴1-1𝐵𝐵𝑉) ∧ (𝐶𝐴𝐶𝑉)) → (𝐹𝐶) ≈ 𝐶)

Proof of Theorem f1imaen2g
StepHypRef Expression
1 simprr 500 . . 3 (((𝐹:𝐴1-1𝐵𝐵𝑉) ∧ (𝐶𝐴𝐶𝑉)) → 𝐶𝑉)
2 simplr 498 . . . 4 (((𝐹:𝐴1-1𝐵𝐵𝑉) ∧ (𝐶𝐴𝐶𝑉)) → 𝐵𝑉)
3 f1f 5229 . . . . . 6 (𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
4 imassrn 4798 . . . . . . 7 (𝐹𝐶) ⊆ ran 𝐹
5 frn 5182 . . . . . . 7 (𝐹:𝐴𝐵 → ran 𝐹𝐵)
64, 5syl5ss 3037 . . . . . 6 (𝐹:𝐴𝐵 → (𝐹𝐶) ⊆ 𝐵)
73, 6syl 14 . . . . 5 (𝐹:𝐴1-1𝐵 → (𝐹𝐶) ⊆ 𝐵)
87ad2antrr 473 . . . 4 (((𝐹:𝐴1-1𝐵𝐵𝑉) ∧ (𝐶𝐴𝐶𝑉)) → (𝐹𝐶) ⊆ 𝐵)
92, 8ssexd 3985 . . 3 (((𝐹:𝐴1-1𝐵𝐵𝑉) ∧ (𝐶𝐴𝐶𝑉)) → (𝐹𝐶) ∈ V)
10 f1ores 5281 . . . 4 ((𝐹:𝐴1-1𝐵𝐶𝐴) → (𝐹𝐶):𝐶1-1-onto→(𝐹𝐶))
1110ad2ant2r 494 . . 3 (((𝐹:𝐴1-1𝐵𝐵𝑉) ∧ (𝐶𝐴𝐶𝑉)) → (𝐹𝐶):𝐶1-1-onto→(𝐹𝐶))
12 f1oen2g 6526 . . 3 ((𝐶𝑉 ∧ (𝐹𝐶) ∈ V ∧ (𝐹𝐶):𝐶1-1-onto→(𝐹𝐶)) → 𝐶 ≈ (𝐹𝐶))
131, 9, 11, 12syl3anc 1175 . 2 (((𝐹:𝐴1-1𝐵𝐵𝑉) ∧ (𝐶𝐴𝐶𝑉)) → 𝐶 ≈ (𝐹𝐶))
1413ensymd 6554 1 (((𝐹:𝐴1-1𝐵𝐵𝑉) ∧ (𝐶𝐴𝐶𝑉)) → (𝐹𝐶) ≈ 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wcel 1439  Vcvv 2620  wss 3000   class class class wbr 3851  ran crn 4453  cres 4454  cima 4455  wf 5024  1-1wf1 5025  1-1-ontowf1o 5027  cen 6509
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-id 4129  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-er 6306  df-en 6512
This theorem is referenced by:  ssenen  6621  phplem4  6625  phplem4dom  6632  phplem4on  6637  fiintim  6693
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