Theorem f1elima 5475

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 5260	. . . 4 ⊢ (F:A–1-1→B → F Fn A)
2		fvelimab 5371	. . . 4 ⊢ ((F Fn A ∧ Y ⊆ A) → ((F ‘X) ∈ (F “ Y) ↔ ∃z ∈ Y (F ‘z) = (F ‘X)))
3	1, 2	sylan 457	. . 3 ⊢ ((F:A–1-1→B ∧ Y ⊆ A) → ((F ‘X) ∈ (F “ Y) ↔ ∃z ∈ Y (F ‘z) = (F ‘X)))
4	3	3adant2 974	. 2 ⊢ ((F:A–1-1→B ∧ X ∈ A ∧ Y ⊆ A) → ((F ‘X) ∈ (F “ Y) ↔ ∃z ∈ Y (F ‘z) = (F ‘X)))
5		ssel 3268	. . . . . . . 8 ⊢ (Y ⊆ A → (z ∈ Y → z ∈ A))
6	5	impac 604	. . . . . . 7 ⊢ ((Y ⊆ A ∧ z ∈ Y) → (z ∈ A ∧ z ∈ Y))
7		f1fveq 5474	. . . . . . . . . . . 12 ⊢ ((F:A–1-1→B ∧ (z ∈ A ∧ X ∈ A)) → ((F ‘z) = (F ‘X) ↔ z = X))
8	7	ancom2s 777	. . . . . . . . . . 11 ⊢ ((F:A–1-1→B ∧ (X ∈ A ∧ z ∈ A)) → ((F ‘z) = (F ‘X) ↔ z = X))
9	8	biimpd 198	. . . . . . . . . 10 ⊢ ((F:A–1-1→B ∧ (X ∈ A ∧ z ∈ A)) → ((F ‘z) = (F ‘X) → z = X))
10	9	anassrs 629	. . . . . . . . 9 ⊢ (((F:A–1-1→B ∧ X ∈ A) ∧ z ∈ A) → ((F ‘z) = (F ‘X) → z = X))
11		eleq1 2413	. . . . . . . . . 10 ⊢ (z = X → (z ∈ Y ↔ X ∈ Y))
12	11	biimpcd 215	. . . . . . . . 9 ⊢ (z ∈ Y → (z = X → X ∈ Y))
13	10, 12	sylan9 638	. . . . . . . 8 ⊢ ((((F:A–1-1→B ∧ X ∈ A) ∧ z ∈ A) ∧ z ∈ Y) → ((F ‘z) = (F ‘X) → X ∈ Y))
14	13	anasss 628	. . . . . . 7 ⊢ (((F:A–1-1→B ∧ X ∈ A) ∧ (z ∈ A ∧ z ∈ Y)) → ((F ‘z) = (F ‘X) → X ∈ Y))
15	6, 14	sylan2 460	. . . . . 6 ⊢ (((F:A–1-1→B ∧ X ∈ A) ∧ (Y ⊆ A ∧ z ∈ Y)) → ((F ‘z) = (F ‘X) → X ∈ Y))
16	15	anassrs 629	. . . . 5 ⊢ ((((F:A–1-1→B ∧ X ∈ A) ∧ Y ⊆ A) ∧ z ∈ Y) → ((F ‘z) = (F ‘X) → X ∈ Y))
17	16	rexlimdva 2739	. . . 4 ⊢ (((F:A–1-1→B ∧ X ∈ A) ∧ Y ⊆ A) → (∃z ∈ Y (F ‘z) = (F ‘X) → X ∈ Y))
18	17	3impa 1146	. . 3 ⊢ ((F:A–1-1→B ∧ X ∈ A ∧ Y ⊆ A) → (∃z ∈ Y (F ‘z) = (F ‘X) → X ∈ Y))
19		eqid 2353	. . . 4 ⊢ (F ‘X) = (F ‘X)
20		fveq2 5329	. . . . . 6 ⊢ (z = X → (F ‘z) = (F ‘X))
21	20	eqeq1d 2361	. . . . 5 ⊢ (z = X → ((F ‘z) = (F ‘X) ↔ (F ‘X) = (F ‘X)))
22	21	rspcev 2956	. . . 4 ⊢ ((X ∈ Y ∧ (F ‘X) = (F ‘X)) → ∃z ∈ Y (F ‘z) = (F ‘X))
23	19, 22	mpan2 652	. . 3 ⊢ (X ∈ Y → ∃z ∈ Y (F ‘z) = (F ‘X))
24	18, 23	impbid1 194	. 2 ⊢ ((F:A–1-1→B ∧ X ∈ A ∧ Y ⊆ A) → (∃z ∈ Y (F ‘z) = (F ‘X) ↔ X ∈ Y))
25	4, 24	bitrd 244	1 ⊢ ((F:A–1-1→B ∧ X ∈ A ∧ Y ⊆ A) → ((F ‘X) ∈ (F “ Y) ↔ X ∈ Y))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  ∃wrex 2616   ⊆ wss 3258   “ cima 4723   Fn wfn 4777  –1-1→wf1 4779   ‘cfv 4782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-ima 4728  df-id 4768  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fv 4796

This theorem is referenced by: (None)