Theorem preimaf1ofi 7262

Description: The preimage of a finite set under a one-to-one, onto function is finite. (Contributed by Jim Kingdon, 24-Sep-2022.)

Hypotheses

Ref	Expression
preimaf1ofi.ss	⊢ (𝜑 → 𝐶 ⊆ 𝐵)
preimaf1ofi.f	⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)
preimaf1ofi.c	⊢ (𝜑 → 𝐶 ∈ Fin)

Assertion

Ref	Expression
preimaf1ofi	⊢ (𝜑 → (◡𝐹 “ 𝐶) ∈ Fin)

Proof of Theorem preimaf1ofi

Step	Hyp	Ref	Expression
1		preimaf1ofi.c	. 2 ⊢ (𝜑 → 𝐶 ∈ Fin)
2		preimaf1ofi.f	. . . 4 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)
3		f1ocnv 5650	. . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴)
4		f1of1 5636	. . . 4 ⊢ (◡𝐹:𝐵–1-1-onto→𝐴 → ◡𝐹:𝐵–1-1→𝐴)
5	2, 3, 4	3syl 17	. . 3 ⊢ (𝜑 → ◡𝐹:𝐵–1-1→𝐴)
6		preimaf1ofi.ss	. . 3 ⊢ (𝜑 → 𝐶 ⊆ 𝐵)
7		f1ores 5652	. . 3 ⊢ ((◡𝐹:𝐵–1-1→𝐴 ∧ 𝐶 ⊆ 𝐵) → (◡𝐹 ↾ 𝐶):𝐶–1-1-onto→(◡𝐹 “ 𝐶))
8	5, 6, 7	syl2anc 415	. 2 ⊢ (𝜑 → (◡𝐹 ↾ 𝐶):𝐶–1-1-onto→(◡𝐹 “ 𝐶))
9		f1ofi 7251	. 2 ⊢ ((𝐶 ∈ Fin ∧ (◡𝐹 ↾ 𝐶):𝐶–1-1-onto→(◡𝐹 “ 𝐶)) → (◡𝐹 “ 𝐶) ∈ Fin)
10	1, 8, 9	syl2anc 415	1 ⊢ (𝜑 → (◡𝐹 “ 𝐶) ∈ Fin)

Syntax hints:   → wi 4   ∈ wcel 2209   ⊆ wss 3220  ^◡ccnv 4771   ↾ cres 4774   “ cima 4775  –1-1→wf1 5372  –1-1-onto→wf1o 5374  Fincfn 7016

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-er 6801  df-en 7017  df-fin 7019

This theorem is referenced by:  fisumss  12142  fprodssdc  12340