Theorem frege121 40685

Description: Lemma for frege122 40686. Proposition 121 of [Frege1879] p. 79. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
frege116.x	⊢ 𝑋 ∈ 𝑈
frege118.y	⊢ 𝑌 ∈ 𝑉
frege120.a	⊢ 𝐴 ∈ 𝑊

Assertion

Ref	Expression
frege121	⊢ ((𝐴 = 𝑋 → 𝑋((t+‘𝑅) ∪ I )𝐴) → (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴 → 𝑋((t+‘𝑅) ∪ I )𝐴))))

Proof of Theorem frege121

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 40684	. 2 ⊢ (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴 → 𝐴 = 𝑋)))
5		frege20 40529	. 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))))

Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111   ∪ cun 3879   class class class wbr 5030   I cid 5424  ^◡ccnv 5518  Fun wfun 6318  ‘cfv 6324  t+ctcl 14336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-frege1 40491  ax-frege2 40492  ax-frege8 40510  ax-frege52a 40558  ax-frege58b 40602

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ifp 1059  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-fun 6326

This theorem is referenced by:  frege122  40686