Theorem frege116 38273

Description: One direction of dffrege115 38272. Proposition 116 of [Frege1879] p. 77. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
frege116.x	⊢ 𝑋 ∈ 𝑈

Assertion

Ref	Expression
frege116	⊢ (Fun ◡◡𝑅 → ∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)))

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

Allowed substitution hints:   𝑈(𝑎,𝑏)

Proof of Theorem frege116

Dummy variable 𝑐 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffrege115 38272	. 2 ⊢ (∀𝑐∀𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ Fun ◡◡𝑅)
2		frege116.x	. . . 4 ⊢ 𝑋 ∈ 𝑈
3	2	frege68c 38225	. . 3 ⊢ ((∀𝑐∀𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ Fun ◡◡𝑅) → (Fun ◡◡𝑅 → [𝑋 / 𝑐]∀𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐))))
4		sbcal 3485	. . . 4 ⊢ ([𝑋 / 𝑐]∀𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ∀𝑏[𝑋 / 𝑐](𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)))
5		sbcimg 3477	. . . . . . 7 ⊢ (𝑋 ∈ 𝑈 → ([𝑋 / 𝑐](𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ([𝑋 / 𝑐]𝑏𝑅𝑐 → [𝑋 / 𝑐]∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐))))
6	2, 5	ax-mp 5	. . . . . 6 ⊢ ([𝑋 / 𝑐](𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ([𝑋 / 𝑐]𝑏𝑅𝑐 → [𝑋 / 𝑐]∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)))
7		sbcbr2g 4710	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝑈 → ([𝑋 / 𝑐]𝑏𝑅𝑐 ↔ 𝑏𝑅⦋𝑋 / 𝑐⦌𝑐))
8	2, 7	ax-mp 5	. . . . . . . 8 ⊢ ([𝑋 / 𝑐]𝑏𝑅𝑐 ↔ 𝑏𝑅⦋𝑋 / 𝑐⦌𝑐)
9		csbvarg 4003	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝑈 → ⦋𝑋 / 𝑐⦌𝑐 = 𝑋)
10	2, 9	ax-mp 5	. . . . . . . . 9 ⊢ ⦋𝑋 / 𝑐⦌𝑐 = 𝑋
11	10	breq2i 4661	. . . . . . . 8 ⊢ (𝑏𝑅⦋𝑋 / 𝑐⦌𝑐 ↔ 𝑏𝑅𝑋)
12	8, 11	bitri 264	. . . . . . 7 ⊢ ([𝑋 / 𝑐]𝑏𝑅𝑐 ↔ 𝑏𝑅𝑋)
13		sbcal 3485	. . . . . . . 8 ⊢ ([𝑋 / 𝑐]∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐) ↔ ∀𝑎[𝑋 / 𝑐](𝑏𝑅𝑎 → 𝑎 = 𝑐))
14		sbcimg 3477	. . . . . . . . . . 11 ⊢ (𝑋 ∈ 𝑈 → ([𝑋 / 𝑐](𝑏𝑅𝑎 → 𝑎 = 𝑐) ↔ ([𝑋 / 𝑐]𝑏𝑅𝑎 → [𝑋 / 𝑐]𝑎 = 𝑐)))
15	2, 14	ax-mp 5	. . . . . . . . . 10 ⊢ ([𝑋 / 𝑐](𝑏𝑅𝑎 → 𝑎 = 𝑐) ↔ ([𝑋 / 𝑐]𝑏𝑅𝑎 → [𝑋 / 𝑐]𝑎 = 𝑐))
16		sbcg 3503	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ 𝑈 → ([𝑋 / 𝑐]𝑏𝑅𝑎 ↔ 𝑏𝑅𝑎))
17	2, 16	ax-mp 5	. . . . . . . . . . 11 ⊢ ([𝑋 / 𝑐]𝑏𝑅𝑎 ↔ 𝑏𝑅𝑎)
18		sbceq2g 3990	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ 𝑈 → ([𝑋 / 𝑐]𝑎 = 𝑐 ↔ 𝑎 = ⦋𝑋 / 𝑐⦌𝑐))
19	2, 18	ax-mp 5	. . . . . . . . . . . 12 ⊢ ([𝑋 / 𝑐]𝑎 = 𝑐 ↔ 𝑎 = ⦋𝑋 / 𝑐⦌𝑐)
20	10	eqeq2i 2634	. . . . . . . . . . . 12 ⊢ (𝑎 = ⦋𝑋 / 𝑐⦌𝑐 ↔ 𝑎 = 𝑋)
21	19, 20	bitri 264	. . . . . . . . . . 11 ⊢ ([𝑋 / 𝑐]𝑎 = 𝑐 ↔ 𝑎 = 𝑋)
22	17, 21	imbi12i 340	. . . . . . . . . 10 ⊢ (([𝑋 / 𝑐]𝑏𝑅𝑎 → [𝑋 / 𝑐]𝑎 = 𝑐) ↔ (𝑏𝑅𝑎 → 𝑎 = 𝑋))
23	15, 22	bitri 264	. . . . . . . . 9 ⊢ ([𝑋 / 𝑐](𝑏𝑅𝑎 → 𝑎 = 𝑐) ↔ (𝑏𝑅𝑎 → 𝑎 = 𝑋))
24	23	albii 1747	. . . . . . . 8 ⊢ (∀𝑎[𝑋 / 𝑐](𝑏𝑅𝑎 → 𝑎 = 𝑐) ↔ ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋))
25	13, 24	bitri 264	. . . . . . 7 ⊢ ([𝑋 / 𝑐]∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐) ↔ ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋))
26	12, 25	imbi12i 340	. . . . . 6 ⊢ (([𝑋 / 𝑐]𝑏𝑅𝑐 → [𝑋 / 𝑐]∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ (𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)))
27	6, 26	bitri 264	. . . . 5 ⊢ ([𝑋 / 𝑐](𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ (𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)))
28	27	albii 1747	. . . 4 ⊢ (∀𝑏[𝑋 / 𝑐](𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)))
29	4, 28	bitri 264	. . 3 ⊢ ([𝑋 / 𝑐]∀𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)))
30	3, 29	syl6ib 241	. 2 ⊢ ((∀𝑐∀𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ Fun ◡◡𝑅) → (Fun ◡◡𝑅 → ∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋))))
31	1, 30	ax-mp 5	1 ⊢ (Fun ◡◡𝑅 → ∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 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:  frege117  38274