Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > fodjuomnilemres | Unicode version |
Description: Lemma for fodjuomni 7014. The final result with expressed as a local definition. (Contributed by Jim Kingdon, 29-Jul-2022.) |
Ref | Expression |
---|---|
fodjuomni.o | Omni |
fodjuomni.fo | ⊔ |
fodjuomni.p | inl |
Ref | Expression |
---|---|
fodjuomnilemres |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq1 5413 | . . . . . 6 | |
2 | 1 | eqeq1d 2146 | . . . . 5 |
3 | 2 | rexbidv 2436 | . . . 4 |
4 | 1 | eqeq1d 2146 | . . . . 5 |
5 | 4 | ralbidv 2435 | . . . 4 |
6 | 3, 5 | orbi12d 782 | . . 3 |
7 | fodjuomni.o | . . . 4 Omni | |
8 | isomnimap 7002 | . . . . 5 Omni Omni | |
9 | 7, 8 | syl 14 | . . . 4 Omni |
10 | 7, 9 | mpbid 146 | . . 3 |
11 | fodjuomni.fo | . . . 4 ⊔ | |
12 | fodjuomni.p | . . . 4 inl | |
13 | 11, 12, 7 | fodjuf 7010 | . . 3 |
14 | 6, 10, 13 | rspcdva 2789 | . 2 |
15 | 11 | adantr 274 | . . . . 5 ⊔ |
16 | simpr 109 | . . . . . 6 | |
17 | fveqeq2 5423 | . . . . . . 7 | |
18 | 17 | cbvrexv 2653 | . . . . . 6 |
19 | 16, 18 | sylib 121 | . . . . 5 |
20 | 15, 12, 19 | fodjum 7011 | . . . 4 |
21 | 20 | ex 114 | . . 3 |
22 | 11 | adantr 274 | . . . . 5 ⊔ |
23 | simpr 109 | . . . . 5 | |
24 | 22, 12, 23 | fodju0 7012 | . . . 4 |
25 | 24 | ex 114 | . . 3 |
26 | 21, 25 | orim12d 775 | . 2 |
27 | 14, 26 | mpd 13 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 697 wceq 1331 wex 1468 wcel 1480 wral 2414 wrex 2415 c0 3358 cif 3469 cmpt 3984 wfo 5116 cfv 5118 (class class class)co 5767 c1o 6299 c2o 6300 cmap 6535 ⊔ cdju 6915 inlcinl 6923 Omnicomni 6997 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-if 3470 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-iord 4283 df-on 4285 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-1o 6306 df-2o 6307 df-map 6537 df-dju 6916 df-inl 6925 df-inr 6926 df-omni 6999 |
This theorem is referenced by: fodjuomni 7014 |
Copyright terms: Public domain | W3C validator |