Theorem sbthlemi8 7236

Description: Lemma for isbth 7239. (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
sbthlemi8	⊢ (((EXMID ∧ Fun ◡𝑓) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → Fun ◡𝐻)

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 sbthlemi8

Step	Hyp	Ref	Expression
1		funres11 5430	. . . 4 ⊢ (Fun ◡𝑓 → Fun ◡(𝑓 ↾ ∪ 𝐷))
2	1	ad2antlr 489	. . 3 ⊢ (((EXMID ∧ Fun ◡𝑓) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → Fun ◡(𝑓 ↾ ∪ 𝐷))
3		funcnvcnv 5417	. . . . . 6 ⊢ (Fun 𝑔 → Fun ◡◡𝑔)
4		funres11 5430	. . . . . 6 ⊢ (Fun ◡◡𝑔 → Fun ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
5	3, 4	syl 14	. . . . 5 ⊢ (Fun 𝑔 → Fun ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
6	5	ad2antrr 488	. . . 4 ⊢ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) → Fun ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
7	6	ad2antrl 490	. . 3 ⊢ (((EXMID ∧ Fun ◡𝑓) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → Fun ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
8		simpll 527	. . . 4 ⊢ (((EXMID ∧ Fun ◡𝑓) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → EXMID)
9		simprll 539	. . . . 5 ⊢ (((EXMID ∧ Fun ◡𝑓) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → (Fun 𝑔 ∧ dom 𝑔 = 𝐵))
10	9	simprd 114	. . . 4 ⊢ (((EXMID ∧ Fun ◡𝑓) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → dom 𝑔 = 𝐵)
11		simprlr 540	. . . 4 ⊢ (((EXMID ∧ Fun ◡𝑓) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → ran 𝑔 ⊆ 𝐴)
12		simprr 533	. . . 4 ⊢ (((EXMID ∧ Fun ◡𝑓) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → Fun ◡𝑔)
13		df-ima 4764	. . . . . . 7 ⊢ (𝑓 “ ∪ 𝐷) = ran (𝑓 ↾ ∪ 𝐷)
14		df-rn 4762	. . . . . . 7 ⊢ ran (𝑓 ↾ ∪ 𝐷) = dom ◡(𝑓 ↾ ∪ 𝐷)
15	13, 14	eqtr2i 2256	. . . . . 6 ⊢ dom ◡(𝑓 ↾ ∪ 𝐷) = (𝑓 “ ∪ 𝐷)
16		df-ima 4764	. . . . . . . 8 ⊢ (◡𝑔 “ (𝐴 ∖ ∪ 𝐷)) = ran (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))
17		df-rn 4762	. . . . . . . 8 ⊢ ran (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)) = dom ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))
18	16, 17	eqtri 2255	. . . . . . 7 ⊢ (◡𝑔 “ (𝐴 ∖ ∪ 𝐷)) = dom ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))
19		sbthlem.1	. . . . . . . 8 ⊢ 𝐴 ∈ V
20		sbthlem.2	. . . . . . . 8 ⊢ 𝐷 = {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓 “ 𝑥))) ⊆ (𝐴 ∖ 𝑥))}
21	19, 20	sbthlemi4 7232	. . . . . . 7 ⊢ ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → (◡𝑔 “ (𝐴 ∖ ∪ 𝐷)) = (𝐵 ∖ (𝑓 “ ∪ 𝐷)))
22	18, 21	eqtr3id 2281	. . . . . 6 ⊢ ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → dom ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)) = (𝐵 ∖ (𝑓 “ ∪ 𝐷)))
23		ineq12 3419	. . . . . 6 ⊢ ((dom ◡(𝑓 ↾ ∪ 𝐷) = (𝑓 “ ∪ 𝐷) ∧ dom ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)) = (𝐵 ∖ (𝑓 “ ∪ 𝐷))) → (dom ◡(𝑓 ↾ ∪ 𝐷) ∩ dom ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) = ((𝑓 “ ∪ 𝐷) ∩ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))
24	15, 22, 23	sylancr 414	. . . . 5 ⊢ ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → (dom ◡(𝑓 ↾ ∪ 𝐷) ∩ dom ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) = ((𝑓 “ ∪ 𝐷) ∩ (𝐵 ∖ (𝑓 “ ∪ 𝐷))))
25		disjdif 3583	. . . . 5 ⊢ ((𝑓 “ ∪ 𝐷) ∩ (𝐵 ∖ (𝑓 “ ∪ 𝐷))) = ∅
26	24, 25	eqtrdi 2283	. . . 4 ⊢ ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔) → (dom ◡(𝑓 ↾ ∪ 𝐷) ∩ dom ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) = ∅)
27	8, 10, 11, 12, 26	syl121anc 1279	. . 3 ⊢ (((EXMID ∧ Fun ◡𝑓) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → (dom ◡(𝑓 ↾ ∪ 𝐷) ∩ dom ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) = ∅)
28		funun 5399	. . 3 ⊢ (((Fun ◡(𝑓 ↾ ∪ 𝐷) ∧ Fun ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) ∧ (dom ◡(𝑓 ↾ ∪ 𝐷) ∩ dom ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) = ∅) → Fun (◡(𝑓 ↾ ∪ 𝐷) ∪ ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))))
29	2, 7, 27, 28	syl21anc 1273	. 2 ⊢ (((EXMID ∧ Fun ◡𝑓) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → Fun (◡(𝑓 ↾ ∪ 𝐷) ∪ ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))))
30		sbthlem.3	. . . . 5 ⊢ 𝐻 = ((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
31	30	cnveqi 4932	. . . 4 ⊢ ◡𝐻 = ◡((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
32		cnvun 5170	. . . 4 ⊢ ◡((𝑓 ↾ ∪ 𝐷) ∪ (◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))) = (◡(𝑓 ↾ ∪ 𝐷) ∪ ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
33	31, 32	eqtri 2255	. . 3 ⊢ ◡𝐻 = (◡(𝑓 ↾ ∪ 𝐷) ∪ ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷)))
34	33	funeqi 5375	. 2 ⊢ (Fun ◡𝐻 ↔ Fun (◡(𝑓 ↾ ∪ 𝐷) ∪ ◡(◡𝑔 ↾ (𝐴 ∖ ∪ 𝐷))))
35	29, 34	sylibr 134	1 ⊢ (((EXMID ∧ Fun ◡𝑓) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔 ⊆ 𝐴) ∧ Fun ◡𝑔)) → Fun ◡𝐻)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1005   = wceq 1398   ∈ wcel 2205  {cab 2220  Vcvv 2815   ∖ cdif 3210   ∪ cun 3211   ∩ cin 3212   ⊆ wss 3213  ∅c0 3510  ∪ cuni 3916  EXMIDwem 4309  ^◡ccnv 4750  dom cdm 4751  ran crn 4752   ↾ cres 4753   “ cima 4754  Fun wfun 5348

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-exmid 4310  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-fun 5356

This theorem is referenced by:  sbthlemi9  7237