Theorem f1elima 5752

Description: Membership in the image of a 1-1 map. (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
f1elima	⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ⊆ 𝐴) → ((𝐹‘𝑋) ∈ (𝐹 “ 𝑌) ↔ 𝑋 ∈ 𝑌))

Proof of Theorem f1elima

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1fn 5405	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴)
2		fvelimab 5552	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ⊆ 𝐴) → ((𝐹‘𝑋) ∈ (𝐹 “ 𝑌) ↔ ∃𝑧 ∈ 𝑌 (𝐹‘𝑧) = (𝐹‘𝑋)))
3	1, 2	sylan 281	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝑌 ⊆ 𝐴) → ((𝐹‘𝑋) ∈ (𝐹 “ 𝑌) ↔ ∃𝑧 ∈ 𝑌 (𝐹‘𝑧) = (𝐹‘𝑋)))
4	3	3adant2 1011	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ⊆ 𝐴) → ((𝐹‘𝑋) ∈ (𝐹 “ 𝑌) ↔ ∃𝑧 ∈ 𝑌 (𝐹‘𝑧) = (𝐹‘𝑋)))
5		ssel 3141	. . . . . . . 8 ⊢ (𝑌 ⊆ 𝐴 → (𝑧 ∈ 𝑌 → 𝑧 ∈ 𝐴))
6	5	impac 379	. . . . . . 7 ⊢ ((𝑌 ⊆ 𝐴 ∧ 𝑧 ∈ 𝑌) → (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝑌))
7		f1fveq 5751	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝑧 ∈ 𝐴 ∧ 𝑋 ∈ 𝐴)) → ((𝐹‘𝑧) = (𝐹‘𝑋) ↔ 𝑧 = 𝑋))
8	7	ancom2s 561	. . . . . . . . . . 11 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝑋 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝐹‘𝑧) = (𝐹‘𝑋) ↔ 𝑧 = 𝑋))
9	8	biimpd 143	. . . . . . . . . 10 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝑋 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝐹‘𝑧) = (𝐹‘𝑋) → 𝑧 = 𝑋))
10	9	anassrs 398	. . . . . . . . 9 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → ((𝐹‘𝑧) = (𝐹‘𝑋) → 𝑧 = 𝑋))
11		eleq1 2233	. . . . . . . . . 10 ⊢ (𝑧 = 𝑋 → (𝑧 ∈ 𝑌 ↔ 𝑋 ∈ 𝑌))
12	11	biimpcd 158	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑌 → (𝑧 = 𝑋 → 𝑋 ∈ 𝑌))
13	10, 12	sylan9 407	. . . . . . . 8 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 ∈ 𝑌) → ((𝐹‘𝑧) = (𝐹‘𝑋) → 𝑋 ∈ 𝑌))
14	13	anasss 397	. . . . . . 7 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) ∧ (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝑌)) → ((𝐹‘𝑧) = (𝐹‘𝑋) → 𝑋 ∈ 𝑌))
15	6, 14	sylan2 284	. . . . . 6 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) ∧ (𝑌 ⊆ 𝐴 ∧ 𝑧 ∈ 𝑌)) → ((𝐹‘𝑧) = (𝐹‘𝑋) → 𝑋 ∈ 𝑌))
16	15	anassrs 398	. . . . 5 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) ∧ 𝑌 ⊆ 𝐴) ∧ 𝑧 ∈ 𝑌) → ((𝐹‘𝑧) = (𝐹‘𝑋) → 𝑋 ∈ 𝑌))
17	16	rexlimdva 2587	. . . 4 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) ∧ 𝑌 ⊆ 𝐴) → (∃𝑧 ∈ 𝑌 (𝐹‘𝑧) = (𝐹‘𝑋) → 𝑋 ∈ 𝑌))
18	17	3impa 1189	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (∃𝑧 ∈ 𝑌 (𝐹‘𝑧) = (𝐹‘𝑋) → 𝑋 ∈ 𝑌))
19		eqid 2170	. . . 4 ⊢ (𝐹‘𝑋) = (𝐹‘𝑋)
20		fveq2 5496	. . . . . 6 ⊢ (𝑧 = 𝑋 → (𝐹‘𝑧) = (𝐹‘𝑋))
21	20	eqeq1d 2179	. . . . 5 ⊢ (𝑧 = 𝑋 → ((𝐹‘𝑧) = (𝐹‘𝑋) ↔ (𝐹‘𝑋) = (𝐹‘𝑋)))
22	21	rspcev 2834	. . . 4 ⊢ ((𝑋 ∈ 𝑌 ∧ (𝐹‘𝑋) = (𝐹‘𝑋)) → ∃𝑧 ∈ 𝑌 (𝐹‘𝑧) = (𝐹‘𝑋))
23	19, 22	mpan2 423	. . 3 ⊢ (𝑋 ∈ 𝑌 → ∃𝑧 ∈ 𝑌 (𝐹‘𝑧) = (𝐹‘𝑋))
24	18, 23	impbid1 141	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (∃𝑧 ∈ 𝑌 (𝐹‘𝑧) = (𝐹‘𝑋) ↔ 𝑋 ∈ 𝑌))
25	4, 24	bitrd 187	1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ⊆ 𝐴) → ((𝐹‘𝑋) ∈ (𝐹 “ 𝑌) ↔ 𝑋 ∈ 𝑌))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 973   = wceq 1348   ∈ wcel 2141  ∃wrex 2449   ⊆ wss 3121   “ cima 4614   Fn wfn 5193  –1-1→wf1 5195  ‘cfv 5198

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fv 5206

This theorem is referenced by:  f1imass  5753  iseqf1olemnab  10444  fprodssdc  11553  ctinfom  12383  ssnnctlemct  12401