Theorem f1elima 7283

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 6805	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴)
2		fvelimab 6981	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ⊆ 𝐴) → ((𝐹‘𝑋) ∈ (𝐹 “ 𝑌) ↔ ∃𝑧 ∈ 𝑌 (𝐹‘𝑧) = (𝐹‘𝑋)))
3	1, 2	sylan 580	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝑌 ⊆ 𝐴) → ((𝐹‘𝑋) ∈ (𝐹 “ 𝑌) ↔ ∃𝑧 ∈ 𝑌 (𝐹‘𝑧) = (𝐹‘𝑋)))
4	3	3adant2 1132	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ⊆ 𝐴) → ((𝐹‘𝑋) ∈ (𝐹 “ 𝑌) ↔ ∃𝑧 ∈ 𝑌 (𝐹‘𝑧) = (𝐹‘𝑋)))
5		ssel 3977	. . . . . . . 8 ⊢ (𝑌 ⊆ 𝐴 → (𝑧 ∈ 𝑌 → 𝑧 ∈ 𝐴))
6	5	impac 552	. . . . . . 7 ⊢ ((𝑌 ⊆ 𝐴 ∧ 𝑧 ∈ 𝑌) → (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝑌))
7		f1fveq 7282	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝑧 ∈ 𝐴 ∧ 𝑋 ∈ 𝐴)) → ((𝐹‘𝑧) = (𝐹‘𝑋) ↔ 𝑧 = 𝑋))
8	7	ancom2s 650	. . . . . . . . . . 11 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝑋 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝐹‘𝑧) = (𝐹‘𝑋) ↔ 𝑧 = 𝑋))
9	8	biimpd 229	. . . . . . . . . 10 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝑋 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝐹‘𝑧) = (𝐹‘𝑋) → 𝑧 = 𝑋))
10	9	anassrs 467	. . . . . . . . 9 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → ((𝐹‘𝑧) = (𝐹‘𝑋) → 𝑧 = 𝑋))
11		eleq1 2829	. . . . . . . . . 10 ⊢ (𝑧 = 𝑋 → (𝑧 ∈ 𝑌 ↔ 𝑋 ∈ 𝑌))
12	11	biimpcd 249	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑌 → (𝑧 = 𝑋 → 𝑋 ∈ 𝑌))
13	10, 12	sylan9 507	. . . . . . . 8 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 ∈ 𝑌) → ((𝐹‘𝑧) = (𝐹‘𝑋) → 𝑋 ∈ 𝑌))
14	13	anasss 466	. . . . . . 7 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) ∧ (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝑌)) → ((𝐹‘𝑧) = (𝐹‘𝑋) → 𝑋 ∈ 𝑌))
15	6, 14	sylan2 593	. . . . . 6 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) ∧ (𝑌 ⊆ 𝐴 ∧ 𝑧 ∈ 𝑌)) → ((𝐹‘𝑧) = (𝐹‘𝑋) → 𝑋 ∈ 𝑌))
16	15	anassrs 467	. . . . 5 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) ∧ 𝑌 ⊆ 𝐴) ∧ 𝑧 ∈ 𝑌) → ((𝐹‘𝑧) = (𝐹‘𝑋) → 𝑋 ∈ 𝑌))
17	16	rexlimdva 3155	. . . 4 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) ∧ 𝑌 ⊆ 𝐴) → (∃𝑧 ∈ 𝑌 (𝐹‘𝑧) = (𝐹‘𝑋) → 𝑋 ∈ 𝑌))
18	17	3impa 1110	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (∃𝑧 ∈ 𝑌 (𝐹‘𝑧) = (𝐹‘𝑋) → 𝑋 ∈ 𝑌))
19		eqid 2737	. . . 4 ⊢ (𝐹‘𝑋) = (𝐹‘𝑋)
20		fveqeq2 6915	. . . . 5 ⊢ (𝑧 = 𝑋 → ((𝐹‘𝑧) = (𝐹‘𝑋) ↔ (𝐹‘𝑋) = (𝐹‘𝑋)))
21	20	rspcev 3622	. . . 4 ⊢ ((𝑋 ∈ 𝑌 ∧ (𝐹‘𝑋) = (𝐹‘𝑋)) → ∃𝑧 ∈ 𝑌 (𝐹‘𝑧) = (𝐹‘𝑋))
22	19, 21	mpan2 691	. . 3 ⊢ (𝑋 ∈ 𝑌 → ∃𝑧 ∈ 𝑌 (𝐹‘𝑧) = (𝐹‘𝑋))
23	18, 22	impbid1 225	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (∃𝑧 ∈ 𝑌 (𝐹‘𝑧) = (𝐹‘𝑋) ↔ 𝑋 ∈ 𝑌))
24	4, 23	bitrd 279	1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ⊆ 𝐴) → ((𝐹‘𝑋) ∈ (𝐹 “ 𝑌) ↔ 𝑋 ∈ 𝑌))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ∃wrex 3070   ⊆ wss 3951   “ cima 5688   Fn wfn 6556  –1-1→wf1 6558  ‘cfv 6561

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fv 6569

This theorem is referenced by:  f1imass  7284  domunfican  9361  acndom2  10094  hashf1lem1  14494  f1omvdconj  19464  gsumzaddlem  19939  lindfmm  21847  axcontlem10  28988  trlsegvdeg  30246  ismtyima  37810