Theorem f1imaen2g 8959

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 8958 does not need ax-rep 5206.) (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 778	. . 3 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ 𝑉)) → 𝐶 ∈ 𝑉)
2		simplr 774	. . . 4 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ 𝑉)) → 𝐵 ∈ 𝑉)
3		f1f 6730	. . . . . 6 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
4		fimass 6682	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 “ 𝐶) ⊆ 𝐵)
5	3, 4	syl 17	. . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → (𝐹 “ 𝐶) ⊆ 𝐵)
6	5	ad2antrr 732	. . . 4 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ 𝑉)) → (𝐹 “ 𝐶) ⊆ 𝐵)
7	2, 6	ssexd 5259	. . 3 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ 𝑉)) → (𝐹 “ 𝐶) ∈ V)
8		f1ores 6788	. . . 4 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶))
9	8	ad2ant2r 753	. . 3 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ 𝑉)) → (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶))
10		f1oen2g 8912	. . 3 ⊢ ((𝐶 ∈ 𝑉 ∧ (𝐹 “ 𝐶) ∈ V ∧ (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶)) → 𝐶 ≈ (𝐹 “ 𝐶))
11	1, 7, 9, 10	syl3anc 1379	. 2 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ 𝑉)) → 𝐶 ≈ (𝐹 “ 𝐶))
12	11	ensymd 8949	1 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ 𝑉)) → (𝐹 “ 𝐶) ≈ 𝐶)

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2119  Vcvv 3432   ⊆ wss 3890   class class class wbr 5079   ↾ cres 5627   “ cima 5628  ⟶wf 6488  –1-1→wf1 6489  –1-1-onto→wf1o 6491   ≈ cen 8887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-er 8640  df-en 8891

This theorem is referenced by:  ssenen  9086  unxpwdom2  9500  znunithash  21546