Theorem frege118 38275

Description: Simplified application of one direction of dffrege115 38272. 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 38215	. . 3 ⊢ (∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)) → [𝑌 / 𝑏](𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)))
3		sbcimg 3477	. . . . 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 4709	. . . . . . 7 ⊢ (𝑌 ∈ 𝑉 → ([𝑌 / 𝑏]𝑏𝑅𝑋 ↔ ⦋𝑌 / 𝑏⦌𝑏𝑅𝑋))
6	1, 5	ax-mp 5	. . . . . 6 ⊢ ([𝑌 / 𝑏]𝑏𝑅𝑋 ↔ ⦋𝑌 / 𝑏⦌𝑏𝑅𝑋)
7		csbvarg 4003	. . . . . . . 8 ⊢ (𝑌 ∈ 𝑉 → ⦋𝑌 / 𝑏⦌𝑏 = 𝑌)
8	1, 7	ax-mp 5	. . . . . . 7 ⊢ ⦋𝑌 / 𝑏⦌𝑏 = 𝑌
9	8	breq1i 4660	. . . . . 6 ⊢ (⦋𝑌 / 𝑏⦌𝑏𝑅𝑋 ↔ 𝑌𝑅𝑋)
10	6, 9	bitri 264	. . . . 5 ⊢ ([𝑌 / 𝑏]𝑏𝑅𝑋 ↔ 𝑌𝑅𝑋)
11		sbcal 3485	. . . . . 6 ⊢ ([𝑌 / 𝑏]∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋) ↔ ∀𝑎[𝑌 / 𝑏](𝑏𝑅𝑎 → 𝑎 = 𝑋))
12		sbcimg 3477	. . . . . . . . 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 4709	. . . . . . . . . . 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 4660	. . . . . . . . . 10 ⊢ (⦋𝑌 / 𝑏⦌𝑏𝑅𝑎 ↔ 𝑌𝑅𝑎)
17	15, 16	bitri 264	. . . . . . . . 9 ⊢ ([𝑌 / 𝑏]𝑏𝑅𝑎 ↔ 𝑌𝑅𝑎)
18		sbcg 3503	. . . . . . . . . 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 340	. . . . . . . 8 ⊢ (([𝑌 / 𝑏]𝑏𝑅𝑎 → [𝑌 / 𝑏]𝑎 = 𝑋) ↔ (𝑌𝑅𝑎 → 𝑎 = 𝑋))
21	13, 20	bitri 264	. . . . . . 7 ⊢ ([𝑌 / 𝑏](𝑏𝑅𝑎 → 𝑎 = 𝑋) ↔ (𝑌𝑅𝑎 → 𝑎 = 𝑋))
22	21	albii 1747	. . . . . 6 ⊢ (∀𝑎[𝑌 / 𝑏](𝑏𝑅𝑎 → 𝑎 = 𝑋) ↔ ∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋))
23	11, 22	bitri 264	. . . . 5 ⊢ ([𝑌 / 𝑏]∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋) ↔ ∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋))
24	10, 23	imbi12i 340	. . . 4 ⊢ (([𝑌 / 𝑏]𝑏𝑅𝑋 → [𝑌 / 𝑏]∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)) ↔ (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋)))
25	4, 24	bitri 264	. . 3 ⊢ ([𝑌 / 𝑏](𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)) ↔ (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋)))
26	2, 25	sylib 208	. 2 ⊢ (∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)) → (𝑌𝑅𝑋 → ∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋)))
27		frege116.x	. . 3 ⊢ 𝑋 ∈ 𝑈
28	27	frege117 38274	. 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)))

Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1481   = wceq 1483   ∈ wcel 1990  [wsbc 3435  ⦋csb 3533   class class class wbr 4653  ^◡ccnv 5113  Fun wfun 5882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-frege1 38084  ax-frege2 38085  ax-frege8 38103  ax-frege52a 38151  ax-frege58b 38195

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-fun 5890

This theorem is referenced by:  frege119  38276