Theorem imaelfm 24077

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 6727	. . . 4 ⊢ (𝐹:𝑌⟶𝑋 → (𝐹 “ 𝑆) ⊆ 𝑋)
2	1	3ad2ant3 1151	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐹 “ 𝑆) ⊆ 𝑋)
3		ssid 3967	. . . 4 ⊢ (𝐹 “ 𝑆) ⊆ (𝐹 “ 𝑆)
4		imaeq2 6059	. . . . . 6 ⊢ (𝑥 = 𝑆 → (𝐹 “ 𝑥) = (𝐹 “ 𝑆))
5	4	sseq1d 3976	. . . . 5 ⊢ (𝑥 = 𝑆 → ((𝐹 “ 𝑥) ⊆ (𝐹 “ 𝑆) ↔ (𝐹 “ 𝑆) ⊆ (𝐹 “ 𝑆)))
6	5	rspcev 3590	. . . 4 ⊢ ((𝑆 ∈ 𝐿 ∧ (𝐹 “ 𝑆) ⊆ (𝐹 “ 𝑆)) → ∃𝑥 ∈ 𝐿 (𝐹 “ 𝑥) ⊆ (𝐹 “ 𝑆))
7	3, 6	mpan2 703	. . 3 ⊢ (𝑆 ∈ 𝐿 → ∃𝑥 ∈ 𝐿 (𝐹 “ 𝑥) ⊆ (𝐹 “ 𝑆))
8	2, 7	anim12i 624	. 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 24074	. . 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 485	. 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 260	1 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑆 ∈ 𝐿) → (𝐹 “ 𝑆) ∈ ((𝑋 FilMap 𝐹)‘𝐵))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∃wrex 3095   ⊆ wss 3913   “ cima 5665  ⟶wf 6533  ‘cfv 6537  (class class class)co 7411  fBascfbas 21479  filGencfg 21480   FilMap cfm 24059

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-fbas 21488  df-fg 21489  df-fm 24064

This theorem is referenced by:  rnelfm  24079  fmfnfmlem2  24081  fmfnfmlem4  24083  fmfnfm  24084  fmco  24087  isfcf  24160  cnextcn  24193