Theorem f1imaeq 5568

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 5567	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) → ((𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷) ↔ 𝐶 ⊆ 𝐷))
2		f1imass 5567	. . . 4 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐷 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴)) → ((𝐹 “ 𝐷) ⊆ (𝐹 “ 𝐶) ↔ 𝐷 ⊆ 𝐶))
3	2	ancom2s 534	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) → ((𝐹 “ 𝐷) ⊆ (𝐹 “ 𝐶) ↔ 𝐷 ⊆ 𝐶))
4	1, 3	anbi12d 458	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) → (((𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷) ∧ (𝐹 “ 𝐷) ⊆ (𝐹 “ 𝐶)) ↔ (𝐶 ⊆ 𝐷 ∧ 𝐷 ⊆ 𝐶)))
5		eqss 3041	. 2 ⊢ ((𝐹 “ 𝐶) = (𝐹 “ 𝐷) ↔ ((𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷) ∧ (𝐹 “ 𝐷) ⊆ (𝐹 “ 𝐶)))
6		eqss 3041	. 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 1290   ⊆ wss 3000   “ cima 4455  –1-1→wf1 5025

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-sbc 2842  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-id 4129  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fv 5036

This theorem is referenced by: (None)