Theorem f1imaeq 7222

Description: Taking images under a one-to-one function preserves equality. (Contributed by Stefan O'Rear, 30-Oct-2014.)

Assertion

Ref	Expression
f1imaeq	⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) → ((𝐹 “ 𝐶) = (𝐹 “ 𝐷) ↔ 𝐶 = 𝐷))

Proof of Theorem f1imaeq

Step	Hyp	Ref	Expression
1		f1imass 7221	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) → ((𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷) ↔ 𝐶 ⊆ 𝐷))
2		f1imass 7221	. . . 4 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐷 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴)) → ((𝐹 “ 𝐷) ⊆ (𝐹 “ 𝐶) ↔ 𝐷 ⊆ 𝐶))
3	2	ancom2s 650	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) → ((𝐹 “ 𝐷) ⊆ (𝐹 “ 𝐶) ↔ 𝐷 ⊆ 𝐶))
4	1, 3	anbi12d 632	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) → (((𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷) ∧ (𝐹 “ 𝐷) ⊆ (𝐹 “ 𝐶)) ↔ (𝐶 ⊆ 𝐷 ∧ 𝐷 ⊆ 𝐶)))
5		eqss 3959	. 2 ⊢ ((𝐹 “ 𝐶) = (𝐹 “ 𝐷) ↔ ((𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷) ∧ (𝐹 “ 𝐷) ⊆ (𝐹 “ 𝐶)))
6		eqss 3959	. 2 ⊢ (𝐶 = 𝐷 ↔ (𝐶 ⊆ 𝐷 ∧ 𝐷 ⊆ 𝐶))
7	4, 5, 6	3bitr4g 314	1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) → ((𝐹 “ 𝐶) = (𝐹 “ 𝐷) ↔ 𝐶 = 𝐷))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ⊆ wss 3911   “ cima 5634  –1-1→wf1 6496

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fv 6507

This theorem is referenced by:  f1imapss  7223  dfac12lem2  10074  hmeoimaf1o  23633  imasf1oxms  24353  isuspgrim0  47867  isuspgrimlem  47868  upgrimtrlslem2  47878  grimedg  47908