![]() |
Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > frege121 | Structured version Visualization version GIF version |
Description: Lemma for frege122 42722. Proposition 121 of [Frege1879] p. 79. (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 |
---|---|
frege121 | ⊢ ((𝐴 = 𝑋 → 𝑋((t+‘𝑅) ∪ I )𝐴) → (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴 → 𝑋((t+‘𝑅) ∪ I )𝐴)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege116.x | . . 3 ⊢ 𝑋 ∈ 𝑈 | |
2 | frege118.y | . . 3 ⊢ 𝑌 ∈ 𝑉 | |
3 | frege120.a | . . 3 ⊢ 𝐴 ∈ 𝑊 | |
4 | 1, 2, 3 | frege120 42720 | . 2 ⊢ (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴 → 𝐴 = 𝑋))) |
5 | frege20 42565 | . 2 ⊢ ((Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴 → 𝐴 = 𝑋))) → ((𝐴 = 𝑋 → 𝑋((t+‘𝑅) ∪ I )𝐴) → (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴 → 𝑋((t+‘𝑅) ∪ I )𝐴))))) | |
6 | 4, 5 | ax-mp 5 | 1 ⊢ ((𝐴 = 𝑋 → 𝑋((t+‘𝑅) ∪ I )𝐴) → (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴 → 𝑋((t+‘𝑅) ∪ I )𝐴)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ∪ cun 3946 class class class wbr 5148 I cid 5573 ◡ccnv 5675 Fun wfun 6535 ‘cfv 6541 t+ctcl 14929 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-frege1 42527 ax-frege2 42528 ax-frege8 42546 ax-frege52a 42594 ax-frege58b 42638 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-ifp 1063 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-fun 6543 |
This theorem is referenced by: frege122 42722 |
Copyright terms: Public domain | W3C validator |