Theorem f1imaeq 5684

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 5683	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) → ((𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷) ↔ 𝐶 ⊆ 𝐷))
2		f1imass 5683	. . . 4 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐷 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴)) → ((𝐹 “ 𝐷) ⊆ (𝐹 “ 𝐶) ↔ 𝐷 ⊆ 𝐶))
3	2	ancom2s 556	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) → ((𝐹 “ 𝐷) ⊆ (𝐹 “ 𝐶) ↔ 𝐷 ⊆ 𝐶))
4	1, 3	anbi12d 465	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) → (((𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷) ∧ (𝐹 “ 𝐷) ⊆ (𝐹 “ 𝐶)) ↔ (𝐶 ⊆ 𝐷 ∧ 𝐷 ⊆ 𝐶)))
5		eqss 3117	. 2 ⊢ ((𝐹 “ 𝐶) = (𝐹 “ 𝐷) ↔ ((𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷) ∧ (𝐹 “ 𝐷) ⊆ (𝐹 “ 𝐶)))
6		eqss 3117	. 2 ⊢ (𝐶 = 𝐷 ↔ (𝐶 ⊆ 𝐷 ∧ 𝐷 ⊆ 𝐶))
7	4, 5, 6	3bitr4g 222	1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) → ((𝐹 “ 𝐶) = (𝐹 “ 𝐷) ↔ 𝐶 = 𝐷))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1332   ⊆ wss 3076   “ cima 4550  –1-1→wf1 5128

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fv 5139

This theorem is referenced by:  hmeoimaf1o  12522