Theorem f1imaeng 8256

Description: A one-to-one function's image under a subset of its domain is equinumerous to the subset. (Contributed by Mario Carneiro, 15-May-2015.)

Assertion

Ref	Expression
f1imaeng	⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ 𝑉) → (𝐹 “ 𝐶) ≈ 𝐶)

Proof of Theorem f1imaeng

Step	Hyp	Ref	Expression
1		f1ores 6371	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶))
2		f1oeng 8215	. . . 4 ⊢ ((𝐶 ∈ 𝑉 ∧ (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶)) → 𝐶 ≈ (𝐹 “ 𝐶))
3	2	ancoms 451	. . 3 ⊢ (((𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶) ∧ 𝐶 ∈ 𝑉) → 𝐶 ≈ (𝐹 “ 𝐶))
4	1, 3	stoic3 1872	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ 𝑉) → 𝐶 ≈ (𝐹 “ 𝐶))
5	4	ensymd 8247	1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ 𝑉) → (𝐹 “ 𝐶) ≈ 𝐶)

Syntax hints:   → wi 4   ∧ w3a 1108   ∈ wcel 2157   ⊆ wss 3770   class class class wbr 4844   ↾ cres 5315   “ cima 5316  –1-1→wf1 6099  –1-1-onto→wf1o 6101   ≈ cen 8193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-rep 4965  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098  ax-un 7184

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3388  df-sbc 3635  df-csb 3730  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-pw 4352  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-iun 4713  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-res 5325  df-ima 5326  df-iota 6065  df-fun 6104  df-fn 6105  df-f 6106  df-f1 6107  df-fo 6108  df-f1o 6109  df-fv 6110  df-er 7983  df-en 8197

This theorem is referenced by:  f1imaen  8258  ackbij1b  9350  enfin1ai  9495  isercolllem2  14736  pmtrfconj  18197  ballotlemro  31100