Theorem sbthlemi4 6954

Description: Lemma for isbth 6961. (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
sbthlemi4	⊢ ((EXMID ∧ (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 sbthlemi4

Step	Hyp	Ref	Expression
1		df-ima 4637	. 2 ⊢ (◡𝑔 “ (𝐴 ∖ ∪ 𝐷)) = ran (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))
2		difss 3261	. . . . . . . 8 ⊢ (𝐵 ∖ (𝑓 “ ∪ 𝐷)) ⊆ 𝐵
3		sseq2 3179	. . . . . . . 8 ⊢ (dom 𝑔 = 𝐵 → ((𝐵 ∖ (𝑓 “ ∪ 𝐷)) ⊆ dom 𝑔 ↔ (𝐵 ∖ (𝑓 “ ∪ 𝐷)) ⊆ 𝐵))
4	2, 3	mpbiri 168	. . . . . . 7 ⊢ (dom 𝑔 = 𝐵 → (𝐵 ∖ (𝑓 “ ∪ 𝐷)) ⊆ dom 𝑔)
5		ssdmres 4926	. . . . . . 7 ⊢ ((𝐵 ∖ (𝑓 “ ∪ 𝐷)) ⊆ dom 𝑔 ↔ dom (𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = (𝐵 ∖ (𝑓 “ ∪ 𝐷)))
6	4, 5	sylib 122	. . . . . 6 ⊢ (dom 𝑔 = 𝐵 → dom (𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = (𝐵 ∖ (𝑓 “ ∪ 𝐷)))
7		dfdm4 4816	. . . . . 6 ⊢ dom (𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = ran ◡(𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))
8	6, 7	eqtr3di 2225	. . . . 5 ⊢ (dom 𝑔 = 𝐵 → (𝐵 ∖ (𝑓 “ ∪ 𝐷)) = ran ◡(𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))
9	8	adantr 276	. . . 4 ⊢ ((dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) → (𝐵 ∖ (𝑓 “ ∪ 𝐷)) = ran ◡(𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))
10	9	3ad2ant2 1019	. . 3 ⊢ ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → (𝐵 ∖ (𝑓 “ ∪ 𝐷)) = ran ◡(𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))
11		funcnvres 5286	. . . . . . 7 ⊢ (Fun ◡𝑔 → ◡(𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = (◡𝑔 ↾ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))))
12	11	3ad2ant3 1020	. . . . . 6 ⊢ ((EXMID ∧ ran 𝑔 ⊆ 𝐴 ∧ Fun ◡𝑔) → ◡(𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = (◡𝑔 ↾ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))))
13		sbthlem.1	. . . . . . . . 9 ⊢ 𝐴 ∈ V
14		sbthlem.2	. . . . . . . . 9 ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))}
15	13, 14	sbthlemi3 6953	. . . . . . . 8 ⊢ ((EXMID ∧ ran 𝑔 ⊆ 𝐴) → (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = (𝐴 ∖ ∪ 𝐷))
16	15	reseq2d 4904	. . . . . . 7 ⊢ ((EXMID ∧ ran 𝑔 ⊆ 𝐴) → (◡𝑔 ↾ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) = (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
17	16	3adant3 1017	. . . . . 6 ⊢ ((EXMID ∧ ran 𝑔 ⊆ 𝐴 ∧ Fun ◡𝑔) → (◡𝑔 ↾ (𝑔 “ (𝐵 ∖ (𝑓 “ ∪ 𝐷)))) = (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
18	12, 17	eqtrd 2210	. . . . 5 ⊢ ((EXMID ∧ ran 𝑔 ⊆ 𝐴 ∧ Fun ◡𝑔) → ◡(𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
19	18	rneqd 4853	. . . 4 ⊢ ((EXMID ∧ ran 𝑔 ⊆ 𝐴 ∧ Fun ◡𝑔) → ran ◡(𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = ran (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
20	19	3adant2l 1232	. . 3 ⊢ ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → ran ◡(𝑔 ↾ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = ran (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
21	10, 20	eqtrd 2210	. 2 ⊢ ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → (𝐵 ∖ (𝑓 “ ∪ 𝐷)) = ran (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
22	1, 21	eqtr4id 2229	1 ⊢ ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → (◡𝑔 “ (𝐴 ∖ ∪ 𝐷)) = (𝐵 ∖ (𝑓 “ ∪ 𝐷)))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  {cab 2163  Vcvv 2737   ∖ cdif 3126   ⊆ wss 3129  ∪ cuni 3808  EXMIDwem 4192  ^◡ccnv 4623  dom cdm 4624  ran crn 4625   ↾ cres 4626   “ cima 4627  Fun wfun 5207

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4119  ax-nul 4127  ax-pow 4172  ax-pr 4207

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3809  df-br 4002  df-opab 4063  df-exmid 4193  df-id 4291  df-xp 4630  df-rel 4631  df-cnv 4632  df-co 4633  df-dm 4634  df-rn 4635  df-res 4636  df-ima 4637  df-fun 5215

This theorem is referenced by:  sbthlemi6  6956  sbthlemi8  6958