Theorem f1elima 7023

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 6578	. . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴)
2		fvelimab 6739	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ⊆ 𝐴) → ((𝐹‘𝑋) ∈ (𝐹 “ 𝑌) ↔ ∃𝑧 ∈ 𝑌 (𝐹‘𝑧) = (𝐹‘𝑋)))
3	1, 2	sylan 582	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝑌 ⊆ 𝐴) → ((𝐹‘𝑋) ∈ (𝐹 “ 𝑌) ↔ ∃𝑧 ∈ 𝑌 (𝐹‘𝑧) = (𝐹‘𝑋)))
4	3	3adant2 1127	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ⊆ 𝐴) → ((𝐹‘𝑋) ∈ (𝐹 “ 𝑌) ↔ ∃𝑧 ∈ 𝑌 (𝐹‘𝑧) = (𝐹‘𝑋)))
5		ssel 3963	. . . . . . . 8 ⊢ (𝑌 ⊆ 𝐴 → (𝑧 ∈ 𝑌 → 𝑧 ∈ 𝐴))
6	5	impac 555	. . . . . . 7 ⊢ ((𝑌 ⊆ 𝐴 ∧ 𝑧 ∈ 𝑌) → (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝑌))
7		f1fveq 7022	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝑧 ∈ 𝐴 ∧ 𝑋 ∈ 𝐴)) → ((𝐹‘𝑧) = (𝐹‘𝑋) ↔ 𝑧 = 𝑋))
8	7	ancom2s 648	. . . . . . . . . . 11 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝑋 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝐹‘𝑧) = (𝐹‘𝑋) ↔ 𝑧 = 𝑋))
9	8	biimpd 231	. . . . . . . . . 10 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝑋 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝐹‘𝑧) = (𝐹‘𝑋) → 𝑧 = 𝑋))
10	9	anassrs 470	. . . . . . . . 9 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) → ((𝐹‘𝑧) = (𝐹‘𝑋) → 𝑧 = 𝑋))
11		eleq1 2902	. . . . . . . . . 10 ⊢ (𝑧 = 𝑋 → (𝑧 ∈ 𝑌 ↔ 𝑋 ∈ 𝑌))
12	11	biimpcd 251	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝑌 → (𝑧 = 𝑋 → 𝑋 ∈ 𝑌))
13	10, 12	sylan9 510	. . . . . . . 8 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 ∈ 𝑌) → ((𝐹‘𝑧) = (𝐹‘𝑋) → 𝑋 ∈ 𝑌))
14	13	anasss 469	. . . . . . 7 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) ∧ (𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝑌)) → ((𝐹‘𝑧) = (𝐹‘𝑋) → 𝑋 ∈ 𝑌))
15	6, 14	sylan2 594	. . . . . 6 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) ∧ (𝑌 ⊆ 𝐴 ∧ 𝑧 ∈ 𝑌)) → ((𝐹‘𝑧) = (𝐹‘𝑋) → 𝑋 ∈ 𝑌))
16	15	anassrs 470	. . . . 5 ⊢ ((((𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) ∧ 𝑌 ⊆ 𝐴) ∧ 𝑧 ∈ 𝑌) → ((𝐹‘𝑧) = (𝐹‘𝑋) → 𝑋 ∈ 𝑌))
17	16	rexlimdva 3286	. . . 4 ⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) ∧ 𝑌 ⊆ 𝐴) → (∃𝑧 ∈ 𝑌 (𝐹‘𝑧) = (𝐹‘𝑋) → 𝑋 ∈ 𝑌))
18	17	3impa 1106	. . 3 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (∃𝑧 ∈ 𝑌 (𝐹‘𝑧) = (𝐹‘𝑋) → 𝑋 ∈ 𝑌))
19		eqid 2823	. . . 4 ⊢ (𝐹‘𝑋) = (𝐹‘𝑋)
20		fveqeq2 6681	. . . . 5 ⊢ (𝑧 = 𝑋 → ((𝐹‘𝑧) = (𝐹‘𝑋) ↔ (𝐹‘𝑋) = (𝐹‘𝑋)))
21	20	rspcev 3625	. . . 4 ⊢ ((𝑋 ∈ 𝑌 ∧ (𝐹‘𝑋) = (𝐹‘𝑋)) → ∃𝑧 ∈ 𝑌 (𝐹‘𝑧) = (𝐹‘𝑋))
22	19, 21	mpan2 689	. . 3 ⊢ (𝑋 ∈ 𝑌 → ∃𝑧 ∈ 𝑌 (𝐹‘𝑧) = (𝐹‘𝑋))
23	18, 22	impbid1 227	. 2 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (∃𝑧 ∈ 𝑌 (𝐹‘𝑧) = (𝐹‘𝑋) ↔ 𝑋 ∈ 𝑌))
24	4, 23	bitrd 281	1 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ⊆ 𝐴) → ((𝐹‘𝑋) ∈ (𝐹 “ 𝑌) ↔ 𝑋 ∈ 𝑌))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∃wrex 3141   ⊆ wss 3938   “ cima 5560   Fn wfn 6352  –1-1→wf1 6354  ‘cfv 6357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fv 6365

This theorem is referenced by:  f1imass  7024  domunfican  8793  acndom2  9482  hashf1lem1  13816  f1omvdconj  18576  gsumzaddlem  19043  lindfmm  20973  axcontlem10  26761  trlsegvdeg  28008  ismtyima  35083