Theorem frege118 45057

Description: Simplified application of one direction of dffrege115 45054. Proposition 118 of [Frege1879] p. 78. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
frege116.x	⊢ 𝑋 ∈ 𝑈
frege118.y	⊢ 𝑌 ∈ 𝑉

Assertion

Ref	Expression
frege118	⊢ (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋)))

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

Allowed substitution hints:   𝑈(𝑎)   𝑉(𝑎)

Proof of Theorem frege118

Dummy variable 𝑏 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frege118.y	. . . 4 ⊢ 𝑌 ∈ 𝑉
2	1	frege58c 44997	. . 3 ⊢ (∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)) → [𝑌 / 𝑏](𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)))
3		sbcimg 3786	. . . . 5 ⊢ (𝑌 ∈ 𝑉 → ([𝑌 / 𝑏](𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)) ↔ ([𝑌 / 𝑏]𝑏𝑅𝑋 → [𝑌 / 𝑏]∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋))))
4	1, 3	ax-mp 5	. . . 4 ⊢ ([𝑌 / 𝑏](𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)) ↔ ([𝑌 / 𝑏]𝑏𝑅𝑋 → [𝑌 / 𝑏]∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)))
5		sbcbr1g 5162	. . . . . . 7 ⊢ (𝑌 ∈ 𝑉 → ([𝑌 / 𝑏]𝑏𝑅𝑋 ↔ ⦋𝑌 / 𝑏⦌𝑏𝑅𝑋))
6	1, 5	ax-mp 5	. . . . . 6 ⊢ ([𝑌 / 𝑏]𝑏𝑅𝑋 ↔ ⦋𝑌 / 𝑏⦌𝑏𝑅𝑋)
7		csbvarg 4391	. . . . . . . 8 ⊢ (𝑌 ∈ 𝑉 → ⦋𝑌 / 𝑏⦌𝑏 = 𝑌)
8	1, 7	ax-mp 5	. . . . . . 7 ⊢ ⦋𝑌 / 𝑏⦌𝑏 = 𝑌
9	8	breq1i 5109	. . . . . 6 ⊢ (⦋𝑌 / 𝑏⦌𝑏𝑅𝑋 ↔ 𝑌𝑅𝑋)
10	6, 9	bitri 278	. . . . 5 ⊢ ([𝑌 / 𝑏]𝑏𝑅𝑋 ↔ 𝑌𝑅𝑋)
11		sbcal 3797	. . . . . 6 ⊢ ([𝑌 / 𝑏]∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋) ↔ ∀𝑎[𝑌 / 𝑏](𝑏𝑅𝑎 → 𝑎 = 𝑋))
12		sbcimg 3786	. . . . . . . . 9 ⊢ (𝑌 ∈ 𝑉 → ([𝑌 / 𝑏](𝑏𝑅𝑎 → 𝑎 = 𝑋) ↔ ([𝑌 / 𝑏]𝑏𝑅𝑎 → [𝑌 / 𝑏]𝑎 = 𝑋)))
13	1, 12	ax-mp 5	. . . . . . . 8 ⊢ ([𝑌 / 𝑏](𝑏𝑅𝑎 → 𝑎 = 𝑋) ↔ ([𝑌 / 𝑏]𝑏𝑅𝑎 → [𝑌 / 𝑏]𝑎 = 𝑋))
14		sbcbr1g 5162	. . . . . . . . . . 11 ⊢ (𝑌 ∈ 𝑉 → ([𝑌 / 𝑏]𝑏𝑅𝑎 ↔ ⦋𝑌 / 𝑏⦌𝑏𝑅𝑎))
15	1, 14	ax-mp 5	. . . . . . . . . 10 ⊢ ([𝑌 / 𝑏]𝑏𝑅𝑎 ↔ ⦋𝑌 / 𝑏⦌𝑏𝑅𝑎)
16	8	breq1i 5109	. . . . . . . . . 10 ⊢ (⦋𝑌 / 𝑏⦌𝑏𝑅𝑎 ↔ 𝑌𝑅𝑎)
17	15, 16	bitri 278	. . . . . . . . 9 ⊢ ([𝑌 / 𝑏]𝑏𝑅𝑎 ↔ 𝑌𝑅𝑎)
18		sbcg 3810	. . . . . . . . . 10 ⊢ (𝑌 ∈ 𝑉 → ([𝑌 / 𝑏]𝑎 = 𝑋 ↔ 𝑎 = 𝑋))
19	1, 18	ax-mp 5	. . . . . . . . 9 ⊢ ([𝑌 / 𝑏]𝑎 = 𝑋 ↔ 𝑎 = 𝑋)
20	17, 19	imbi12i 353	. . . . . . . 8 ⊢ (([𝑌 / 𝑏]𝑏𝑅𝑎 → [𝑌 / 𝑏]𝑎 = 𝑋) ↔ (𝑌𝑅𝑎 → 𝑎 = 𝑋))
21	13, 20	bitri 278	. . . . . . 7 ⊢ ([𝑌 / 𝑏](𝑏𝑅𝑎 → 𝑎 = 𝑋) ↔ (𝑌𝑅𝑎 → 𝑎 = 𝑋))
22	21	albii 1851	. . . . . 6 ⊢ (∀𝑎[𝑌 / 𝑏](𝑏𝑅𝑎 → 𝑎 = 𝑋) ↔ ∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋))
23	11, 22	bitri 278	. . . . 5 ⊢ ([𝑌 / 𝑏]∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋) ↔ ∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋))
24	10, 23	imbi12i 353	. . . 4 ⊢ (([𝑌 / 𝑏]𝑏𝑅𝑋 → [𝑌 / 𝑏]∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)) ↔ (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋)))
25	4, 24	bitri 278	. . 3 ⊢ ([𝑌 / 𝑏](𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)) ↔ (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋)))
26	2, 25	sylib 221	. 2 ⊢ (∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)) → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋)))
27		frege116.x	. . 3 ⊢ 𝑋 ∈ 𝑈
28	27	frege117 45056	. 2 ⊢ ((∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)) → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋))) → (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋))))
29	26, 28	ax-mp 5	1 ⊢ (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1567   = wceq 1569   ∈ wcel 2144  [wsbc 3738  ⦋csb 3846   class class class wbr 5102  ^◡ccnv 5649  Fun wfun 6532

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390  ax-frege1 44866  ax-frege2 44867  ax-frege8 44885  ax-frege52a 44933  ax-frege58b 44977

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ifp 1079  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-fun 6540

This theorem is used by:  frege119  45058