Theorem f1elima 5794

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 5442	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴)
2		fvelimab 5592	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ⊆ 𝐴) → ((𝐹‘𝑋) ∈ (𝐹 “ 𝑌) ↔ ∃𝑧 ∈ 𝑌 (𝐹‘𝑧) = (𝐹‘𝑋)))
3	1, 2	sylan 283	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝑌 ⊆ 𝐴) → ((𝐹‘𝑋) ∈ (𝐹 “ 𝑌) ↔ ∃𝑧 ∈ 𝑌 (𝐹‘𝑧) = (𝐹‘𝑋)))
4	3	3adant2 1018	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ⊆ 𝐴) → ((𝐹‘𝑋) ∈ (𝐹 “ 𝑌) ↔ ∃𝑧 ∈ 𝑌 (𝐹‘𝑧) = (𝐹‘𝑋)))
5		ssel 3164	. . . . . . . 8 ⊢ (𝑌 ⊆ 𝐴 → (𝑧 ∈ 𝑌 → 𝑧 ∈ 𝐴))
6	5	impac 381	. . . . . . 7 ⊢ ((𝑌 ⊆ 𝐴 ∧ 𝑧 ∈ 𝑌) → (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝑌))
7		f1fveq 5793	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝑧 ∈ 𝐴 ∧ 𝑋 ∈ 𝐴)) → ((𝐹‘𝑧) = (𝐹‘𝑋) ↔ 𝑧 = 𝑋))
8	7	ancom2s 566	. . . . . . . . . . 11 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝑋 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝐹‘𝑧) = (𝐹‘𝑋) ↔ 𝑧 = 𝑋))
9	8	biimpd 144	. . . . . . . . . 10 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝑋 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝐹‘𝑧) = (𝐹‘𝑋) → 𝑧 = 𝑋))
10	9	anassrs 400	. . . . . . . . 9 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → ((𝐹‘𝑧) = (𝐹‘𝑋) → 𝑧 = 𝑋))
11		eleq1 2252	. . . . . . . . . 10 ⊢ (𝑧 = 𝑋 → (𝑧 ∈ 𝑌 ↔ 𝑋 ∈ 𝑌))
12	11	biimpcd 159	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑌 → (𝑧 = 𝑋 → 𝑋 ∈ 𝑌))
13	10, 12	sylan9 409	. . . . . . . 8 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 ∈ 𝑌) → ((𝐹‘𝑧) = (𝐹‘𝑋) → 𝑋 ∈ 𝑌))
14	13	anasss 399	. . . . . . 7 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) ∧ (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝑌)) → ((𝐹‘𝑧) = (𝐹‘𝑋) → 𝑋 ∈ 𝑌))
15	6, 14	sylan2 286	. . . . . 6 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) ∧ (𝑌 ⊆ 𝐴 ∧ 𝑧 ∈ 𝑌)) → ((𝐹‘𝑧) = (𝐹‘𝑋) → 𝑋 ∈ 𝑌))
16	15	anassrs 400	. . . . 5 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) ∧ 𝑌 ⊆ 𝐴) ∧ 𝑧 ∈ 𝑌) → ((𝐹‘𝑧) = (𝐹‘𝑋) → 𝑋 ∈ 𝑌))
17	16	rexlimdva 2607	. . . 4 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) ∧ 𝑌 ⊆ 𝐴) → (∃𝑧 ∈ 𝑌 (𝐹‘𝑧) = (𝐹‘𝑋) → 𝑋 ∈ 𝑌))
18	17	3impa 1196	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (∃𝑧 ∈ 𝑌 (𝐹‘𝑧) = (𝐹‘𝑋) → 𝑋 ∈ 𝑌))
19		eqid 2189	. . . 4 ⊢ (𝐹‘𝑋) = (𝐹‘𝑋)
20		fveq2 5534	. . . . . 6 ⊢ (𝑧 = 𝑋 → (𝐹‘𝑧) = (𝐹‘𝑋))
21	20	eqeq1d 2198	. . . . 5 ⊢ (𝑧 = 𝑋 → ((𝐹‘𝑧) = (𝐹‘𝑋) ↔ (𝐹‘𝑋) = (𝐹‘𝑋)))
22	21	rspcev 2856	. . . 4 ⊢ ((𝑋 ∈ 𝑌 ∧ (𝐹‘𝑋) = (𝐹‘𝑋)) → ∃𝑧 ∈ 𝑌 (𝐹‘𝑧) = (𝐹‘𝑋))
23	19, 22	mpan2 425	. . 3 ⊢ (𝑋 ∈ 𝑌 → ∃𝑧 ∈ 𝑌 (𝐹‘𝑧) = (𝐹‘𝑋))
24	18, 23	impbid1 142	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (∃𝑧 ∈ 𝑌 (𝐹‘𝑧) = (𝐹‘𝑋) ↔ 𝑋 ∈ 𝑌))
25	4, 24	bitrd 188	1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ⊆ 𝐴) → ((𝐹‘𝑋) ∈ (𝐹 “ 𝑌) ↔ 𝑋 ∈ 𝑌))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364   ∈ wcel 2160  ∃wrex 2469   ⊆ wss 3144   “ cima 4647   Fn wfn 5230  –1-1→wf1 5232  ‘cfv 5235

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fv 5243

This theorem is referenced by:  f1imass  5795  iseqf1olemnab  10518  fprodssdc  11629  ctinfom  12478  ssnnctlemct  12496