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Theorem f1imaen2g 6771
Description: A one-to-one function's image under a subset of its domain is equinumerous to the subset. (This version of f1imaen 6772 does not need ax-setind 4521.) (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 527 . . 3 (((𝐹:𝐴1-1𝐵𝐵𝑉) ∧ (𝐶𝐴𝐶𝑉)) → 𝐶𝑉)
2 simplr 525 . . . 4 (((𝐹:𝐴1-1𝐵𝐵𝑉) ∧ (𝐶𝐴𝐶𝑉)) → 𝐵𝑉)
3 f1f 5403 . . . . . 6 (𝐹:𝐴1-1𝐵𝐹:𝐴𝐵)
4 imassrn 4964 . . . . . . 7 (𝐹𝐶) ⊆ ran 𝐹
5 frn 5356 . . . . . . 7 (𝐹:𝐴𝐵 → ran 𝐹𝐵)
64, 5sstrid 3158 . . . . . 6 (𝐹:𝐴𝐵 → (𝐹𝐶) ⊆ 𝐵)
73, 6syl 14 . . . . 5 (𝐹:𝐴1-1𝐵 → (𝐹𝐶) ⊆ 𝐵)
87ad2antrr 485 . . . 4 (((𝐹:𝐴1-1𝐵𝐵𝑉) ∧ (𝐶𝐴𝐶𝑉)) → (𝐹𝐶) ⊆ 𝐵)
92, 8ssexd 4129 . . 3 (((𝐹:𝐴1-1𝐵𝐵𝑉) ∧ (𝐶𝐴𝐶𝑉)) → (𝐹𝐶) ∈ V)
10 f1ores 5457 . . . 4 ((𝐹:𝐴1-1𝐵𝐶𝐴) → (𝐹𝐶):𝐶1-1-onto→(𝐹𝐶))
1110ad2ant2r 506 . . 3 (((𝐹:𝐴1-1𝐵𝐵𝑉) ∧ (𝐶𝐴𝐶𝑉)) → (𝐹𝐶):𝐶1-1-onto→(𝐹𝐶))
12 f1oen2g 6733 . . 3 ((𝐶𝑉 ∧ (𝐹𝐶) ∈ V ∧ (𝐹𝐶):𝐶1-1-onto→(𝐹𝐶)) → 𝐶 ≈ (𝐹𝐶))
131, 9, 11, 12syl3anc 1233 . 2 (((𝐹:𝐴1-1𝐵𝐵𝑉) ∧ (𝐶𝐴𝐶𝑉)) → 𝐶 ≈ (𝐹𝐶))
1413ensymd 6761 1 (((𝐹:𝐴1-1𝐵𝐵𝑉) ∧ (𝐶𝐴𝐶𝑉)) → (𝐹𝐶) ≈ 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wcel 2141  Vcvv 2730  wss 3121   class class class wbr 3989  ran crn 4612  cres 4613  cima 4614  wf 5194  1-1wf1 5195  1-1-ontowf1o 5197  cen 6716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-er 6513  df-en 6719
This theorem is referenced by:  ssenen  6829  phplem4  6833  phplem4dom  6840  phplem4on  6845  fiintim  6906
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