Theorem sbthlem4 9127

Description: Lemma for sbth 9134. (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 5697	. 2 ⊢ (◡𝑔 “ (𝐴 ∖ ∪ 𝐷)) = ran (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))
2		difss 4139	. . . . . . 7 ⊢ (𝐵 ∖ (𝑓 “ ∪ 𝐷)) ⊆ 𝐵
3		sseq2 4017	. . . . . . 7 ⊢ (dom 𝑔 = 𝐵 → ((𝐵 ∖ (𝑓 “ ∪ 𝐷)) ⊆ dom 𝑔 ↔ (𝐵 ∖ (𝑓 “ ∪ 𝐷)) ⊆ 𝐵))
4	2, 3	mpbiri 257	. . . . . 6 ⊢ (dom 𝑔 = 𝐵 → (𝐵 ∖ (𝑓 “ ∪ 𝐷)) ⊆ dom 𝑔)
5		ssdmres 6025	. . . . . 6 ⊢ ((𝐵 ∖ (𝑓 “ ∪ 𝐷)) ⊆ dom 𝑔 ↔ dom (𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = (𝐵 ∖ (𝑓 “ ∪ 𝐷)))
6	4, 5	sylib 217	. . . . 5 ⊢ (dom 𝑔 = 𝐵 → dom (𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = (𝐵 ∖ (𝑓 “ ∪ 𝐷)))
7		dfdm4 5904	. . . . 5 ⊢ dom (𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = ran ◡(𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))
8	6, 7	eqtr3di 2784	. . . 4 ⊢ (dom 𝑔 = 𝐵 → (𝐵 ∖ (𝑓 “ ∪ 𝐷)) = ran ◡(𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))
9		funcnvres 6639	. . . . . 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 9126	. . . . . . 7 ⊢ (ran 𝑔 ⊆ 𝐴 → (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = (𝐴 ∖ ∪ 𝐷))
13	12	reseq2d 5992	. . . . . 6 ⊢ (ran 𝑔 ⊆ 𝐴 → (◡𝑔 ↾ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) = (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
14	9, 13	sylan9eqr 2791	. . . . 5 ⊢ ((ran 𝑔 ⊆ 𝐴 ∧ Fun ◡𝑔) → ◡(𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
15	14	rneqd 5946	. . . 4 ⊢ ((ran 𝑔 ⊆ 𝐴 ∧ Fun ◡𝑔) → ran ◡(𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = ran (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
16	8, 15	sylan9eq 2789	. . 3 ⊢ ((dom 𝑔 = 𝐵 ∧ (ran 𝑔 ⊆ 𝐴 ∧ Fun ◡𝑔)) → (𝐵 ∖ (𝑓 “ ∪ 𝐷)) = ran (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
17	16	anassrs 466	. 2 ⊢ (((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → (𝐵 ∖ (𝑓 “ ∪ 𝐷)) = ran (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
18	1, 17	eqtr4id 2788	1 ⊢ (((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → (◡𝑔 “ (𝐴 ∖ ∪ 𝐷)) = (𝐵 ∖ (𝑓 “ ∪ 𝐷)))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1534   ∈ wcel 2100  {cab 2706  Vcvv 3472   ∖ cdif 3955   ⊆ wss 3958  ∪ cuni 4916  ^◡ccnv 5683  dom cdm 5684  ran crn 5685   ↾ cres 5686   “ cima 5687  Fun wfun 6550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2102  ax-9 2110  ax-10 2133  ax-12 2170  ax-ext 2700  ax-sep 5305  ax-nul 5312  ax-pr 5435

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2062  df-mo 2532  df-eu 2561  df-clab 2707  df-cleq 2721  df-clel 2806  df-ral 3055  df-rex 3064  df-rab 3429  df-v 3474  df-dif 3961  df-un 3963  df-in 3965  df-ss 3975  df-nul 4334  df-if 4535  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4917  df-br 5155  df-opab 5217  df-id 5582  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-fun 6558

This theorem is referenced by:  sbthlem6  9129  sbthlem8  9131