Theorem frege116 39113

Description: One direction of dffrege115 39112. 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 39112	. 2 ⊢ (∀𝑐∀𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ Fun ◡◡𝑅)
2		frege116.x	. . . 4 ⊢ 𝑋 ∈ 𝑈
3	2	frege68c 39065	. . 3 ⊢ ((∀𝑐∀𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ Fun ◡◡𝑅) → (Fun ◡◡𝑅 → [𝑋 / 𝑐]∀𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐))))
4		sbcal 3712	. . . 4 ⊢ ([𝑋 / 𝑐]∀𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ∀𝑏[𝑋 / 𝑐](𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)))
5		sbcimg 3704	. . . . . . 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 4931	. . . . . . . . 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 4227	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝑈 → ⦋𝑋 / 𝑐⦌𝑐 = 𝑋)
10	2, 9	ax-mp 5	. . . . . . . . 9 ⊢ ⦋𝑋 / 𝑐⦌𝑐 = 𝑋
11	10	breq2i 4881	. . . . . . . 8 ⊢ (𝑏𝑅⦋𝑋 / 𝑐⦌𝑐 ↔ 𝑏𝑅𝑋)
12	8, 11	bitri 267	. . . . . . 7 ⊢ ([𝑋 / 𝑐]𝑏𝑅𝑐 ↔ 𝑏𝑅𝑋)
13		sbcal 3712	. . . . . . . 8 ⊢ ([𝑋 / 𝑐]∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐) ↔ ∀𝑎[𝑋 / 𝑐](𝑏𝑅𝑎 → 𝑎 = 𝑐))
14		sbcimg 3704	. . . . . . . . . . 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 3728	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ 𝑈 → ([𝑋 / 𝑐]𝑏𝑅𝑎 ↔ 𝑏𝑅𝑎))
17	2, 16	ax-mp 5	. . . . . . . . . . 11 ⊢ ([𝑋 / 𝑐]𝑏𝑅𝑎 ↔ 𝑏𝑅𝑎)
18		sbceq2g 4214	. . . . . . . . . . . . 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 2837	. . . . . . . . . . . 12 ⊢ (𝑎 = ⦋𝑋 / 𝑐⦌𝑐 ↔ 𝑎 = 𝑋)
21	19, 20	bitri 267	. . . . . . . . . . 11 ⊢ ([𝑋 / 𝑐]𝑎 = 𝑐 ↔ 𝑎 = 𝑋)
22	17, 21	imbi12i 342	. . . . . . . . . 10 ⊢ (([𝑋 / 𝑐]𝑏𝑅𝑎 → [𝑋 / 𝑐]𝑎 = 𝑐) ↔ (𝑏𝑅𝑎 → 𝑎 = 𝑋))
23	15, 22	bitri 267	. . . . . . . . 9 ⊢ ([𝑋 / 𝑐](𝑏𝑅𝑎 → 𝑎 = 𝑐) ↔ (𝑏𝑅𝑎 → 𝑎 = 𝑋))
24	23	albii 1920	. . . . . . . 8 ⊢ (∀𝑎[𝑋 / 𝑐](𝑏𝑅𝑎 → 𝑎 = 𝑐) ↔ ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋))
25	13, 24	bitri 267	. . . . . . 7 ⊢ ([𝑋 / 𝑐]∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐) ↔ ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋))
26	12, 25	imbi12i 342	. . . . . 6 ⊢ (([𝑋 / 𝑐]𝑏𝑅𝑐 → [𝑋 / 𝑐]∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ (𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)))
27	6, 26	bitri 267	. . . . 5 ⊢ ([𝑋 / 𝑐](𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ (𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)))
28	27	albii 1920	. . . 4 ⊢ (∀𝑏[𝑋 / 𝑐](𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)))
29	4, 28	bitri 267	. . 3 ⊢ ([𝑋 / 𝑐]∀𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)))
30	3, 29	syl6ib 243	. 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 198  ∀wal 1656   = wceq 1658   ∈ wcel 2166  [wsbc 3662  ⦋csb 3757   class class class wbr 4873  ^◡ccnv 5341  Fun wfun 6117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127  ax-frege1 38924  ax-frege2 38925  ax-frege8 38943  ax-frege52a 38991  ax-frege58b 39035

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-ifp 1092  df-3an 1115  df-tru 1662  df-fal 1672  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4874  df-opab 4936  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-fun 6125

This theorem is referenced by:  frege117  39114