Theorem frege117 43289

Description: Lemma for frege118 43290. Proposition 117 of [Frege1879] p. 78. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
frege116.x	⊢ 𝑋 ∈ 𝑈

Assertion

Ref	Expression
frege117	⊢ ((∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)) → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋))) → (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋))))

Distinct variable groups:   𝑎,𝑏,𝑅   𝑋,𝑎,𝑏

Allowed substitution hints:   𝑈(𝑎,𝑏)   𝑌(𝑎,𝑏)

Proof of Theorem frege117

Step	Hyp	Ref	Expression
1		frege116.x	. . 3 ⊢ 𝑋 ∈ 𝑈
2	1	frege116 43288	. 2 ⊢ (Fun ◡◡𝑅 → ∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)))
3		frege9 43121	. 2 ⊢ ((Fun ◡◡𝑅 → ∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋))) → ((∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)) → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋))) → (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋)))))
4	2, 3	ax-mp 5	1 ⊢ ((∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)) → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋))) → (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋))))

Syntax hints:   → wi 4  ∀wal 1531   = wceq 1533   ∈ wcel 2098   class class class wbr 5141  ^◡ccnv 5668  Fun wfun 6530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-frege1 43099  ax-frege2 43100  ax-frege8 43118  ax-frege52a 43166  ax-frege58b 43210

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ifp 1060  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-fun 6538

This theorem is referenced by:  frege118  43290