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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 40315. 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 40258 | . . 3 ⊢ (∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋) → [𝐴 / 𝑎](𝑌𝑅𝑎 → 𝑎 = 𝑋)) |
3 | sbcim1 3823 | . . . 4 ⊢ ([𝐴 / 𝑎](𝑌𝑅𝑎 → 𝑎 = 𝑋) → ([𝐴 / 𝑎]𝑌𝑅𝑎 → [𝐴 / 𝑎]𝑎 = 𝑋)) | |
4 | sbcbr2g 5115 | . . . . . 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 4381 | . . . . . . 7 ⊢ (𝐴 ∈ 𝑊 → ⦋𝐴 / 𝑎⦌𝑎 = 𝐴) | |
7 | 1, 6 | ax-mp 5 | . . . . . 6 ⊢ ⦋𝐴 / 𝑎⦌𝑎 = 𝐴 |
8 | 7 | breq2i 5065 | . . . . 5 ⊢ (𝑌𝑅⦋𝐴 / 𝑎⦌𝑎 ↔ 𝑌𝑅𝐴) |
9 | 5, 8 | bitri 277 | . . . 4 ⊢ ([𝐴 / 𝑎]𝑌𝑅𝑎 ↔ 𝑌𝑅𝐴) |
10 | sbceq1g 4364 | . . . . . 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 2824 | . . . . 5 ⊢ (⦋𝐴 / 𝑎⦌𝑎 = 𝑋 ↔ 𝐴 = 𝑋) |
13 | 11, 12 | bitri 277 | . . . 4 ⊢ ([𝐴 / 𝑎]𝑎 = 𝑋 ↔ 𝐴 = 𝑋) |
14 | 3, 9, 13 | 3imtr3g 297 | . . 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 40319 | . 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 208 ∀wal 1529 = wceq 1531 ∈ wcel 2108 [wsbc 3770 ⦋csb 3881 class class class wbr 5057 ◡ccnv 5547 Fun wfun 6342 |
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 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-frege1 40127 ax-frege2 40128 ax-frege8 40146 ax-frege52a 40194 ax-frege58b 40238 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ifp 1058 df-3an 1084 df-tru 1534 df-fal 1544 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ral 3141 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-fun 6350 |
This theorem is referenced by: frege121 40321 |
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