Theorem sbthlem6 8318

Description: Lemma for sbth 8323. (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))}
sbthlem.3	⊢ 𝐻 = ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))

Assertion

Ref	Expression
sbthlem6	⊢ ((ran 𝑓 ⊆ 𝐵 ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → ran 𝐻 = 𝐵)

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

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

Proof of Theorem sbthlem6

Step	Hyp	Ref	Expression
1		df-ima 5326	. . . . 5 ⊢ (◡𝑔 “ (𝐴 ∖ ∪ 𝐷)) = ran (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))
2		sbthlem.1	. . . . . 6 ⊢ 𝐴 ∈ V
3		sbthlem.2	. . . . . 6 ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))}
4	2, 3	sbthlem4 8316	. . . . 5 ⊢ (((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → (◡𝑔 “ (𝐴 ∖ ∪ 𝐷)) = (𝐵 ∖ (𝑓 “ ∪ 𝐷)))
5	1, 4	syl5reqr 2849	. . . 4 ⊢ (((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → (𝐵 ∖ (𝑓 “ ∪ 𝐷)) = ran (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
6	5	uneq2d 3966	. . 3 ⊢ (((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → ((𝑓 “ ∪ 𝐷) ∪ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = ((𝑓 “ ∪ 𝐷) ∪ ran (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))))
7		rnun 5759	. . . 4 ⊢ ran ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) = (ran (𝑓 ↾ ∪ 𝐷) ∪ ran (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
8		sbthlem.3	. . . . 5 ⊢ 𝐻 = ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
9	8	rneqi 5556	. . . 4 ⊢ ran 𝐻 = ran ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
10		df-ima 5326	. . . . 5 ⊢ (𝑓 “ ∪ 𝐷) = ran (𝑓 ↾ ∪ 𝐷)
11	10	uneq1i 3962	. . . 4 ⊢ ((𝑓 “ ∪ 𝐷) ∪ ran (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) = (ran (𝑓 ↾ ∪ 𝐷) ∪ ran (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
12	7, 9, 11	3eqtr4i 2832	. . 3 ⊢ ran 𝐻 = ((𝑓 “ ∪ 𝐷) ∪ ran (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
13	6, 12	syl6reqr 2853	. 2 ⊢ (((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → ran 𝐻 = ((𝑓 “ ∪ 𝐷) ∪ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))
14		imassrn 5695	. . . 4 ⊢ (𝑓 “ ∪ 𝐷) ⊆ ran 𝑓
15		sstr2 3806	. . . 4 ⊢ ((𝑓 “ ∪ 𝐷) ⊆ ran 𝑓 → (ran 𝑓 ⊆ 𝐵 → (𝑓 “ ∪ 𝐷) ⊆ 𝐵))
16	14, 15	ax-mp 5	. . 3 ⊢ (ran 𝑓 ⊆ 𝐵 → (𝑓 “ ∪ 𝐷) ⊆ 𝐵)
17		undif 4244	. . 3 ⊢ ((𝑓 “ ∪ 𝐷) ⊆ 𝐵 ↔ ((𝑓 “ ∪ 𝐷) ∪ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = 𝐵)
18	16, 17	sylib 210	. 2 ⊢ (ran 𝑓 ⊆ 𝐵 → ((𝑓 “ ∪ 𝐷) ∪ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = 𝐵)
19	13, 18	sylan9eqr 2856	1 ⊢ ((ran 𝑓 ⊆ 𝐵 ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → ran 𝐻 = 𝐵)

Syntax hints:   → wi 4   ∧ wa 385   = wceq 1653   ∈ wcel 2157  {cab 2786  Vcvv 3386   ∖ cdif 3767   ∪ cun 3768   ⊆ wss 3770  ∪ cuni 4629  ^◡ccnv 5312  dom cdm 5313  ran crn 5314   ↾ cres 5315   “ cima 5316  Fun wfun 6096

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pr 5098

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-br 4845  df-opab 4907  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-res 5325  df-ima 5326  df-fun 6104

This theorem is referenced by:  sbthlem9  8321