Theorem sbthlem4 9088

Description: Lemma for sbth 9095. (Contributed by NM, 27-Mar-1998.)

Hypotheses

Ref	Expression
sbthlem.1	⊢ 𝐴 ∈ V
sbthlem.2	⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))}

Assertion

Ref	Expression
sbthlem4	⊢ (((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → (◡𝑔 “ (𝐴 ∖ ∪ 𝐷)) = (𝐵 ∖ (𝑓 “ ∪ 𝐷)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐷   𝑥,𝑓   𝑥,𝑔

Allowed substitution hints:   𝐴(𝑓, 𝑔)   𝐵(𝑓, 𝑔)   𝐷(𝑓, 𝑔)

Proof of Theorem sbthlem4

Step	Hyp	Ref	Expression
1		df-ima 5668	. 2 ⊢ (◡𝑔 “ (𝐴 ∖ ∪ 𝐷)) = ran (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))
2		difss 4083	. . . . . . 7 ⊢ (𝐵 ∖ (𝑓 “ ∪ 𝐷)) ⊆ 𝐵
3		sseq2 3957	. . . . . . 7 ⊢ (dom 𝑔 = 𝐵 → ((𝐵 ∖ (𝑓 “ ∪ 𝐷)) ⊆ dom 𝑔 ↔ (𝐵 ∖ (𝑓 “ ∪ 𝐷)) ⊆ 𝐵))
4	2, 3	mpbiri 261	. . . . . 6 ⊢ (dom 𝑔 = 𝐵 → (𝐵 ∖ (𝑓 “ ∪ 𝐷)) ⊆ dom 𝑔)
5		ssdmres 6006	. . . . . 6 ⊢ ((𝐵 ∖ (𝑓 “ ∪ 𝐷)) ⊆ dom 𝑔 ↔ dom (𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = (𝐵 ∖ (𝑓 “ ∪ 𝐷)))
6	4, 5	sylib 221	. . . . 5 ⊢ (dom 𝑔 = 𝐵 → dom (𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = (𝐵 ∖ (𝑓 “ ∪ 𝐷)))
7		dfdm4 5879	. . . . 5 ⊢ dom (𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = ran ◡(𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))
8	6, 7	eqtr3di 2810	. . . 4 ⊢ (dom 𝑔 = 𝐵 → (𝐵 ∖ (𝑓 “ ∪ 𝐷)) = ran ◡(𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))
9		funcnvres 6611	. . . . . 6 ⊢ (Fun ◡𝑔 → ◡(𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = (◡𝑔 ↾ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))))
10		sbthlem.1	. . . . . . . 8 ⊢ 𝐴 ∈ V
11		sbthlem.2	. . . . . . . 8 ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))}
12	10, 11	sbthlem3 9087	. . . . . . 7 ⊢ (ran 𝑔 ⊆ 𝐴 → (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = (𝐴 ∖ ∪ 𝐷))
13	12	reseq2d 5972	. . . . . 6 ⊢ (ran 𝑔 ⊆ 𝐴 → (◡𝑔 ↾ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) = (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
14	9, 13	sylan9eqr 2817	. . . . 5 ⊢ ((ran 𝑔 ⊆ 𝐴 ∧ Fun ◡𝑔) → ◡(𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
15	14	rneqd 5922	. . . 4 ⊢ ((ran 𝑔 ⊆ 𝐴 ∧ Fun ◡𝑔) → ran ◡(𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = ran (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
16	8, 15	sylan9eq 2815	. . 3 ⊢ ((dom 𝑔 = 𝐵 ∧ (ran 𝑔 ⊆ 𝐴 ∧ Fun ◡𝑔)) → (𝐵 ∖ (𝑓 “ ∪ 𝐷)) = ran (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
17	16	anassrs 473	. 2 ⊢ (((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → (𝐵 ∖ (𝑓 “ ∪ 𝐷)) = ran (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
18	1, 17	eqtr4id 2814	1 ⊢ (((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → (◡𝑔 “ (𝐴 ∖ ∪ 𝐷)) = (𝐵 ∖ (𝑓 “ ∪ 𝐷)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2738  Vcvv 3450   ∖ cdif 3896   ⊆ wss 3899  ∪ cuni 4867  ^◡ccnv 5654  dom cdm 5655  ran crn 5656   ↾ cres 5657   “ cima 5658  Fun wfun 6527

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-fun 6535

This theorem is used by:  sbthlem6  9090  sbthlem8  9092