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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege120 | Structured version Visualization version GIF version |
Description: Simplified application of one direction of dffrege115 43645. Proposition 120 of [Frege1879] p. 78. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege116.x | ⊢ 𝑋 ∈ 𝑈 |
frege118.y | ⊢ 𝑌 ∈ 𝑉 |
frege120.a | ⊢ 𝐴 ∈ 𝑊 |
Ref | Expression |
---|---|
frege120 | ⊢ (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴 → 𝐴 = 𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege120.a | . . . 4 ⊢ 𝐴 ∈ 𝑊 | |
2 | 1 | frege58c 43588 | . . 3 ⊢ (∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋) → [𝐴 / 𝑎](𝑌𝑅𝑎 → 𝑎 = 𝑋)) |
3 | sbcim1 3833 | . . . 4 ⊢ ([𝐴 / 𝑎](𝑌𝑅𝑎 → 𝑎 = 𝑋) → ([𝐴 / 𝑎]𝑌𝑅𝑎 → [𝐴 / 𝑎]𝑎 = 𝑋)) | |
4 | sbcbr2g 5211 | . . . . . 6 ⊢ (𝐴 ∈ 𝑊 → ([𝐴 / 𝑎]𝑌𝑅𝑎 ↔ 𝑌𝑅⦋𝐴 / 𝑎⦌𝑎)) | |
5 | 1, 4 | ax-mp 5 | . . . . 5 ⊢ ([𝐴 / 𝑎]𝑌𝑅𝑎 ↔ 𝑌𝑅⦋𝐴 / 𝑎⦌𝑎) |
6 | csbvarg 4436 | . . . . . . 7 ⊢ (𝐴 ∈ 𝑊 → ⦋𝐴 / 𝑎⦌𝑎 = 𝐴) | |
7 | 1, 6 | ax-mp 5 | . . . . . 6 ⊢ ⦋𝐴 / 𝑎⦌𝑎 = 𝐴 |
8 | 7 | breq2i 5161 | . . . . 5 ⊢ (𝑌𝑅⦋𝐴 / 𝑎⦌𝑎 ↔ 𝑌𝑅𝐴) |
9 | 5, 8 | bitri 274 | . . . 4 ⊢ ([𝐴 / 𝑎]𝑌𝑅𝑎 ↔ 𝑌𝑅𝐴) |
10 | sbceq1g 4419 | . . . . . 6 ⊢ (𝐴 ∈ 𝑊 → ([𝐴 / 𝑎]𝑎 = 𝑋 ↔ ⦋𝐴 / 𝑎⦌𝑎 = 𝑋)) | |
11 | 1, 10 | ax-mp 5 | . . . . 5 ⊢ ([𝐴 / 𝑎]𝑎 = 𝑋 ↔ ⦋𝐴 / 𝑎⦌𝑎 = 𝑋) |
12 | 7 | eqeq1i 2731 | . . . . 5 ⊢ (⦋𝐴 / 𝑎⦌𝑎 = 𝑋 ↔ 𝐴 = 𝑋) |
13 | 11, 12 | bitri 274 | . . . 4 ⊢ ([𝐴 / 𝑎]𝑎 = 𝑋 ↔ 𝐴 = 𝑋) |
14 | 3, 9, 13 | 3imtr3g 294 | . . 3 ⊢ ([𝐴 / 𝑎](𝑌𝑅𝑎 → 𝑎 = 𝑋) → (𝑌𝑅𝐴 → 𝐴 = 𝑋)) |
15 | 2, 14 | syl 17 | . 2 ⊢ (∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋) → (𝑌𝑅𝐴 → 𝐴 = 𝑋)) |
16 | frege116.x | . . 3 ⊢ 𝑋 ∈ 𝑈 | |
17 | frege118.y | . . 3 ⊢ 𝑌 ∈ 𝑉 | |
18 | 16, 17 | frege119 43649 | . 2 ⊢ ((∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋) → (𝑌𝑅𝐴 → 𝐴 = 𝑋)) → (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴 → 𝐴 = 𝑋)))) |
19 | 15, 18 | ax-mp 5 | 1 ⊢ (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴 → 𝐴 = 𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1532 = wceq 1534 ∈ wcel 2099 [wsbc 3776 ⦋csb 3892 class class class wbr 5153 ◡ccnv 5681 Fun wfun 6548 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 ax-frege1 43457 ax-frege2 43458 ax-frege8 43476 ax-frege52a 43524 ax-frege58b 43568 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-ifp 1061 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-br 5154 df-opab 5216 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-fun 6556 |
This theorem is referenced by: frege121 43651 |
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