frege123 - Mathbox for Richard Penner

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Theorem frege123 44169

Description: Lemma for frege124 44170. Proposition 123 of [Frege1879] p. 79. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)

**Hypotheses**
Ref	Expression
frege123.x	⊢ 𝑋 ∈ 𝑈
frege123.y	⊢ 𝑌 ∈ 𝑉

**Assertion**
Ref	Expression
frege123	⊢ ((∀𝑎(𝑌𝑅𝑎 → 𝑋((t+‘𝑅) ∪ I )𝑎) → (𝑌(t+‘𝑅)𝑀 → 𝑋((t+‘𝑅) ∪ I )𝑀)) → (Fun ^◡^◡𝑅 → (𝑌𝑅𝑋 → (𝑌(t+‘𝑅)𝑀 → 𝑋((t+‘𝑅) ∪ I )𝑀))))

Distinct variable groups: 𝑅,𝑎 𝑋,𝑎 𝑌,𝑎 Allowed substitution hints: 𝑈(𝑎) 𝑀(𝑎) 𝑉(𝑎)

**Proof of Theorem frege123**
Step	Hyp	Ref	Expression
1		frege123.x	. . . 4 ⊢ 𝑋 ∈ 𝑈
2		frege123.y	. . . 4 ⊢ 𝑌 ∈ 𝑉
3		vex 3442	. . . 4 ⊢ 𝑎 ∈ V
4	1, 2, 3	frege122 44168	. . 3 ⊢ (Fun ^◡^◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝑎 → 𝑋((t+‘𝑅) ∪ I )𝑎)))
5	4	alrimdv 1930	. 2 ⊢ (Fun ^◡^◡𝑅 → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎 → 𝑋((t+‘𝑅) ∪ I )𝑎)))
6		frege19 44007	. 2 ⊢ ((Fun ^◡^◡𝑅 → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎 → 𝑋((t+‘𝑅) ∪ I )𝑎))) → ((∀𝑎(𝑌𝑅𝑎 → 𝑋((t+‘𝑅) ∪ I )𝑎) → (𝑌(t+‘𝑅)𝑀 → 𝑋((t+‘𝑅) ∪ I )𝑀)) → (Fun ^◡^◡𝑅 → (𝑌𝑅𝑋 → (𝑌(t+‘𝑅)𝑀 → 𝑋((t+‘𝑅) ∪ I )𝑀)))))
7	5, 6	ax-mp 5	1 ⊢ ((∀𝑎(𝑌𝑅𝑎 → 𝑋((t+‘𝑅) ∪ I )𝑎) → (𝑌(t+‘𝑅)𝑀 → 𝑋((t+‘𝑅) ∪ I )𝑀)) → (Fun ^◡^◡𝑅 → (𝑌𝑅𝑋 → (𝑌(t+‘𝑅)𝑀 → 𝑋((t+‘𝑅) ∪ I )𝑀))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∀wal 1539 ∈ wcel 2113 Vcvv 3438 ∪ cun 3897 class class class wbr 5096 I cid 5516 ^◡ccnv 5621 Fun wfun 6484 ‘cfv 6490 t+ctcl 14906

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-sep 5239 ax-nul 5249 ax-pr 5375 ax-frege1 43973 ax-frege2 43974 ax-frege8 43992 ax-frege52a 44040 ax-frege58b 44084

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ifp 1063 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ral 3050 df-rex 3059 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-ss 3916 df-nul 4284 df-if 4478 df-sn 4579 df-pr 4581 df-op 4585 df-br 5097 df-opab 5159 df-id 5517 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-fun 6492

This theorem is referenced by: frege124 44170

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