Theorem frege116 40680

Description: One direction of dffrege115 40679. 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 40679	. 2 ⊢ (∀𝑐∀𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ Fun ◡◡𝑅)
2		frege116.x	. . . 4 ⊢ 𝑋 ∈ 𝑈
3	2	frege68c 40632	. . 3 ⊢ ((∀𝑐∀𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ Fun ◡◡𝑅) → (Fun ◡◡𝑅 → [𝑋 / 𝑐]∀𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐))))
4		sbcal 3780	. . . 4 ⊢ ([𝑋 / 𝑐]∀𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ∀𝑏[𝑋 / 𝑐](𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)))
5		sbcimg 3767	. . . . . . 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 5088	. . . . . . . . 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 4339	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝑈 → ⦋𝑋 / 𝑐⦌𝑐 = 𝑋)
10	2, 9	ax-mp 5	. . . . . . . . 9 ⊢ ⦋𝑋 / 𝑐⦌𝑐 = 𝑋
11	10	breq2i 5038	. . . . . . . 8 ⊢ (𝑏𝑅⦋𝑋 / 𝑐⦌𝑐 ↔ 𝑏𝑅𝑋)
12	8, 11	bitri 278	. . . . . . 7 ⊢ ([𝑋 / 𝑐]𝑏𝑅𝑐 ↔ 𝑏𝑅𝑋)
13		sbcal 3780	. . . . . . . 8 ⊢ ([𝑋 / 𝑐]∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐) ↔ ∀𝑎[𝑋 / 𝑐](𝑏𝑅𝑎 → 𝑎 = 𝑐))
14		sbcimg 3767	. . . . . . . . . . 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 3793	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ 𝑈 → ([𝑋 / 𝑐]𝑏𝑅𝑎 ↔ 𝑏𝑅𝑎))
17	2, 16	ax-mp 5	. . . . . . . . . . 11 ⊢ ([𝑋 / 𝑐]𝑏𝑅𝑎 ↔ 𝑏𝑅𝑎)
18		sbceq2g 4324	. . . . . . . . . . . . 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 2811	. . . . . . . . . . . 12 ⊢ (𝑎 = ⦋𝑋 / 𝑐⦌𝑐 ↔ 𝑎 = 𝑋)
21	19, 20	bitri 278	. . . . . . . . . . 11 ⊢ ([𝑋 / 𝑐]𝑎 = 𝑐 ↔ 𝑎 = 𝑋)
22	17, 21	imbi12i 354	. . . . . . . . . 10 ⊢ (([𝑋 / 𝑐]𝑏𝑅𝑎 → [𝑋 / 𝑐]𝑎 = 𝑐) ↔ (𝑏𝑅𝑎 → 𝑎 = 𝑋))
23	15, 22	bitri 278	. . . . . . . . 9 ⊢ ([𝑋 / 𝑐](𝑏𝑅𝑎 → 𝑎 = 𝑐) ↔ (𝑏𝑅𝑎 → 𝑎 = 𝑋))
24	23	albii 1821	. . . . . . . 8 ⊢ (∀𝑎[𝑋 / 𝑐](𝑏𝑅𝑎 → 𝑎 = 𝑐) ↔ ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋))
25	13, 24	bitri 278	. . . . . . 7 ⊢ ([𝑋 / 𝑐]∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐) ↔ ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋))
26	12, 25	imbi12i 354	. . . . . 6 ⊢ (([𝑋 / 𝑐]𝑏𝑅𝑐 → [𝑋 / 𝑐]∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ (𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)))
27	6, 26	bitri 278	. . . . 5 ⊢ ([𝑋 / 𝑐](𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ (𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)))
28	27	albii 1821	. . . 4 ⊢ (∀𝑏[𝑋 / 𝑐](𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)))
29	4, 28	bitri 278	. . 3 ⊢ ([𝑋 / 𝑐]∀𝑏(𝑏𝑅𝑐 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑐)) ↔ ∀𝑏(𝑏𝑅𝑋 → ∀𝑎(𝑏𝑅𝑎 → 𝑎 = 𝑋)))
30	3, 29	syl6ib 254	. 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 209  ∀wal 1536   = wceq 1538   ∈ wcel 2111  [wsbc 3720  ⦋csb 3828   class class class wbr 5030  ^◡ccnv 5518  Fun wfun 6318

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-frege1 40491  ax-frege2 40492  ax-frege8 40510  ax-frege52a 40558  ax-frege58b 40602

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ifp 1059  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-fun 6326

This theorem is referenced by:  frege117  40681