Theorem imaelfm 23959

Description: An image of a filter element is in the image filter. (Contributed by Jeff Hankins, 5-Oct-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Hypothesis

Ref	Expression
imaelfm.l	⊢ 𝐿 = (𝑌filGen𝐵)

Assertion

Ref	Expression
imaelfm	⊢ (((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑆 ∈ 𝐿) → (𝐹 “ 𝑆) ∈ ((𝑋 FilMap 𝐹)‘𝐵))

Proof of Theorem imaelfm

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fimass 6756	. . . 4 ⊢ (𝐹:𝑌⟶𝑋 → (𝐹 “ 𝑆) ⊆ 𝑋)
2	1	3ad2ant3 1136	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐹 “ 𝑆) ⊆ 𝑋)
3		ssid 4006	. . . 4 ⊢ (𝐹 “ 𝑆) ⊆ (𝐹 “ 𝑆)
4		imaeq2 6074	. . . . . 6 ⊢ (𝑥 = 𝑆 → (𝐹 “ 𝑥) = (𝐹 “ 𝑆))
5	4	sseq1d 4015	. . . . 5 ⊢ (𝑥 = 𝑆 → ((𝐹 “ 𝑥) ⊆ (𝐹 “ 𝑆) ↔ (𝐹 “ 𝑆) ⊆ (𝐹 “ 𝑆)))
6	5	rspcev 3622	. . . 4 ⊢ ((𝑆 ∈ 𝐿 ∧ (𝐹 “ 𝑆) ⊆ (𝐹 “ 𝑆)) → ∃𝑥 ∈ 𝐿 (𝐹 “ 𝑥) ⊆ (𝐹 “ 𝑆))
7	3, 6	mpan2 691	. . 3 ⊢ (𝑆 ∈ 𝐿 → ∃𝑥 ∈ 𝐿 (𝐹 “ 𝑥) ⊆ (𝐹 “ 𝑆))
8	2, 7	anim12i 613	. 2 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑆 ∈ 𝐿) → ((𝐹 “ 𝑆) ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐿 (𝐹 “ 𝑥) ⊆ (𝐹 “ 𝑆)))
9		imaelfm.l	. . . 4 ⊢ 𝐿 = (𝑌filGen𝐵)
10	9	elfm2 23956	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝐹 “ 𝑆) ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ ((𝐹 “ 𝑆) ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐿 (𝐹 “ 𝑥) ⊆ (𝐹 “ 𝑆))))
11	10	adantr 480	. 2 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑆 ∈ 𝐿) → ((𝐹 “ 𝑆) ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ ((𝐹 “ 𝑆) ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐿 (𝐹 “ 𝑥) ⊆ (𝐹 “ 𝑆))))
12	8, 11	mpbird 257	1 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑆 ∈ 𝐿) → (𝐹 “ 𝑆) ∈ ((𝑋 FilMap 𝐹)‘𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ∃wrex 3070   ⊆ wss 3951   “ cima 5688  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431  fBascfbas 21352  filGencfg 21353   FilMap cfm 23941

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-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  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-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  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-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-fbas 21361  df-fg 21362  df-fm 23946

This theorem is referenced by:  rnelfm  23961  fmfnfmlem2  23963  fmfnfmlem4  23965  fmfnfm  23966  fmco  23969  isfcf  24042  cnextcn  24075