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Mirrors > Home > NFE Home > Th. List > sbthlem2 | GIF version |
Description: Lemma for sbth 6206. Eliminate hypotheses from sbthlem1 6203. Theorem XI.1.14 of [Rosser] p. 350. (Contributed by SF, 10-Mar-2015.) |
Ref | Expression |
---|---|
sbthlem2.1 | ⊢ R ∈ V |
Ref | Expression |
---|---|
sbthlem2 | ⊢ (((Fun R ∧ Fun ◡R) ∧ (B ∈ V ∧ B ⊆ dom R ∧ ran R ⊆ B)) → ran R ≈ B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseq1 3292 | . . . . . 6 ⊢ (b = B → (b ⊆ dom R ↔ B ⊆ dom R)) | |
2 | sseq2 3293 | . . . . . 6 ⊢ (b = B → (ran R ⊆ b ↔ ran R ⊆ B)) | |
3 | 1, 2 | anbi12d 691 | . . . . 5 ⊢ (b = B → ((b ⊆ dom R ∧ ran R ⊆ b) ↔ (B ⊆ dom R ∧ ran R ⊆ B))) |
4 | breq2 4643 | . . . . . 6 ⊢ (b = B → (ran R ≈ b ↔ ran R ≈ B)) | |
5 | 4 | imbi2d 307 | . . . . 5 ⊢ (b = B → (((Fun R ∧ Fun ◡R) → ran R ≈ b) ↔ ((Fun R ∧ Fun ◡R) → ran R ≈ B))) |
6 | 3, 5 | imbi12d 311 | . . . 4 ⊢ (b = B → (((b ⊆ dom R ∧ ran R ⊆ b) → ((Fun R ∧ Fun ◡R) → ran R ≈ b)) ↔ ((B ⊆ dom R ∧ ran R ⊆ B) → ((Fun R ∧ Fun ◡R) → ran R ≈ B)))) |
7 | sbthlem2.1 | . . . . . 6 ⊢ R ∈ V | |
8 | vex 2862 | . . . . . 6 ⊢ b ∈ V | |
9 | eqid 2353 | . . . . . 6 ⊢ Clos1 ((b ∖ ran R), R) = Clos1 ((b ∖ ran R), R) | |
10 | eqid 2353 | . . . . . 6 ⊢ (b ∩ Clos1 ((b ∖ ran R), R)) = (b ∩ Clos1 ((b ∖ ran R), R)) | |
11 | eqid 2353 | . . . . . 6 ⊢ (b ∖ Clos1 ((b ∖ ran R), R)) = (b ∖ Clos1 ((b ∖ ran R), R)) | |
12 | eqid 2353 | . . . . . 6 ⊢ (ran R ∩ Clos1 ((b ∖ ran R), R)) = (ran R ∩ Clos1 ((b ∖ ran R), R)) | |
13 | eqid 2353 | . . . . . 6 ⊢ (ran R ∖ Clos1 ((b ∖ ran R), R)) = (ran R ∖ Clos1 ((b ∖ ran R), R)) | |
14 | 7, 8, 9, 10, 11, 12, 13 | sbthlem1 6203 | . . . . 5 ⊢ (((Fun R ∧ Fun ◡R) ∧ (b ⊆ dom R ∧ ran R ⊆ b)) → ran R ≈ b) |
15 | 14 | expcom 424 | . . . 4 ⊢ ((b ⊆ dom R ∧ ran R ⊆ b) → ((Fun R ∧ Fun ◡R) → ran R ≈ b)) |
16 | 6, 15 | vtoclg 2914 | . . 3 ⊢ (B ∈ V → ((B ⊆ dom R ∧ ran R ⊆ B) → ((Fun R ∧ Fun ◡R) → ran R ≈ B))) |
17 | 16 | 3impib 1149 | . 2 ⊢ ((B ∈ V ∧ B ⊆ dom R ∧ ran R ⊆ B) → ((Fun R ∧ Fun ◡R) → ran R ≈ B)) |
18 | 17 | impcom 419 | 1 ⊢ (((Fun R ∧ Fun ◡R) ∧ (B ∈ V ∧ B ⊆ dom R ∧ ran R ⊆ B)) → ran R ≈ B) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 ∧ w3a 934 = wceq 1642 ∈ wcel 1710 Vcvv 2859 ∖ cdif 3206 ∩ cin 3208 ⊆ wss 3257 class class class wbr 4639 ◡ccnv 4771 dom cdm 4772 ran crn 4773 Fun wfun 4775 Clos1 cclos1 5872 ≈ cen 6028 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-1st 4723 df-swap 4724 df-sset 4725 df-co 4726 df-ima 4727 df-si 4728 df-id 4767 df-xp 4784 df-cnv 4785 df-rn 4786 df-dm 4787 df-res 4788 df-fun 4789 df-fn 4790 df-f 4791 df-f1 4792 df-fo 4793 df-f1o 4794 df-2nd 4797 df-txp 5736 df-fix 5740 df-ins2 5750 df-ins3 5752 df-image 5754 df-clos1 5873 df-en 6029 |
This theorem is referenced by: sbthlem3 6205 |
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