frege123 - Mathbox for Richard Penner

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Theorem frege123 41212

Description: Lemma for frege124 41213. 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 3402	. . . 4 ⊢ 𝑎 ∈ V
4	1, 2, 3	frege122 41211	. . 3 ⊢ (Fun ^◡^◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝑎 → 𝑋((t+‘𝑅) ∪ I )𝑎)))
5	4	alrimdv 1937	. 2 ⊢ (Fun ^◡^◡𝑅 → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎 → 𝑋((t+‘𝑅) ∪ I )𝑎)))
6		frege19 41050	. 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 1541 ∈ wcel 2112 Vcvv 3398 ∪ cun 3851 class class class wbr 5039 I cid 5439 ^◡ccnv 5535 Fun wfun 6352 ‘cfv 6358 t+ctcl 14513

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-frege1 41016 ax-frege2 41017 ax-frege8 41035 ax-frege52a 41083 ax-frege58b 41127

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-ifp 1064 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-fun 6360

This theorem is referenced by: frege124 41213

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