Theorem f1imaen2g 9010

Description: If a function is one-to-one, then the image of a subset of its domain under it is equinumerous to the subset. (This version of f1imaeng 9009 does not need ax-rep 5285.) (Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario Carneiro, 25-Jun-2015.)

Assertion

Ref	Expression
f1imaen2g	⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ 𝑉)) → (𝐹 “ 𝐶) ≈ 𝐶)

Proof of Theorem f1imaen2g

Step	Hyp	Ref	Expression
1		simprr 771	. . 3 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ 𝑉)) → 𝐶 ∈ 𝑉)
2		simplr 767	. . . 4 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ 𝑉)) → 𝐵 ∈ 𝑉)
3		f1f 6787	. . . . . 6 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
4		fimass 6738	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 “ 𝐶) ⊆ 𝐵)
5	3, 4	syl 17	. . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → (𝐹 “ 𝐶) ⊆ 𝐵)
6	5	ad2antrr 724	. . . 4 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ 𝑉)) → (𝐹 “ 𝐶) ⊆ 𝐵)
7	2, 6	ssexd 5324	. . 3 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ 𝑉)) → (𝐹 “ 𝐶) ∈ V)
8		f1ores 6847	. . . 4 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶))
9	8	ad2ant2r 745	. . 3 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ 𝑉)) → (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶))
10		f1oen2g 8963	. . 3 ⊢ ((𝐶 ∈ 𝑉 ∧ (𝐹 “ 𝐶) ∈ V ∧ (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶)) → 𝐶 ≈ (𝐹 “ 𝐶))
11	1, 7, 9, 10	syl3anc 1371	. 2 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ 𝑉)) → 𝐶 ≈ (𝐹 “ 𝐶))
12	11	ensymd 9000	1 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ 𝑉)) → (𝐹 “ 𝐶) ≈ 𝐶)

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2106  Vcvv 3474   ⊆ wss 3948   class class class wbr 5148   ↾ cres 5678   “ cima 5679  ⟶wf 6539  –1-1→wf1 6540  –1-1-onto→wf1o 6542   ≈ cen 8935

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-er 8702  df-en 8939

This theorem is referenced by:  ssenen  9150  phplem4OLD  9219  fiint  9323  unxpwdom2  9582  znunithash  21119