Theorem frege120 42719

Description: Simplified application of one direction of dffrege115 42714. Proposition 120 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
frege120.a	⊢ 𝐴 ∈ 𝑊

Assertion

Ref	Expression
frege120	⊢ (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴 → 𝐴 = 𝑋)))

Proof of Theorem frege120

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frege120.a	. . . 4 ⊢ 𝐴 ∈ 𝑊
2	1	frege58c 42657	. . 3 ⊢ (∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋) → [𝐴 / 𝑎](𝑌𝑅𝑎 → 𝑎 = 𝑋))
3		sbcim1 3832	. . . 4 ⊢ ([𝐴 / 𝑎](𝑌𝑅𝑎 → 𝑎 = 𝑋) → ([𝐴 / 𝑎]𝑌𝑅𝑎 → [𝐴 / 𝑎]𝑎 = 𝑋))
4		sbcbr2g 5205	. . . . . 6 ⊢ (𝐴 ∈ 𝑊 → ([𝐴 / 𝑎]𝑌𝑅𝑎 ↔ 𝑌𝑅⦋𝐴 / 𝑎⦌𝑎))
5	1, 4	ax-mp 5	. . . . 5 ⊢ ([𝐴 / 𝑎]𝑌𝑅𝑎 ↔ 𝑌𝑅⦋𝐴 / 𝑎⦌𝑎)
6		csbvarg 4430	. . . . . . 7 ⊢ (𝐴 ∈ 𝑊 → ⦋𝐴 / 𝑎⦌𝑎 = 𝐴)
7	1, 6	ax-mp 5	. . . . . 6 ⊢ ⦋𝐴 / 𝑎⦌𝑎 = 𝐴
8	7	breq2i 5155	. . . . 5 ⊢ (𝑌𝑅⦋𝐴 / 𝑎⦌𝑎 ↔ 𝑌𝑅𝐴)
9	5, 8	bitri 274	. . . 4 ⊢ ([𝐴 / 𝑎]𝑌𝑅𝑎 ↔ 𝑌𝑅𝐴)
10		sbceq1g 4413	. . . . . 6 ⊢ (𝐴 ∈ 𝑊 → ([𝐴 / 𝑎]𝑎 = 𝑋 ↔ ⦋𝐴 / 𝑎⦌𝑎 = 𝑋))
11	1, 10	ax-mp 5	. . . . 5 ⊢ ([𝐴 / 𝑎]𝑎 = 𝑋 ↔ ⦋𝐴 / 𝑎⦌𝑎 = 𝑋)
12	7	eqeq1i 2737	. . . . 5 ⊢ (⦋𝐴 / 𝑎⦌𝑎 = 𝑋 ↔ 𝐴 = 𝑋)
13	11, 12	bitri 274	. . . 4 ⊢ ([𝐴 / 𝑎]𝑎 = 𝑋 ↔ 𝐴 = 𝑋)
14	3, 9, 13	3imtr3g 294	. . 3 ⊢ ([𝐴 / 𝑎](𝑌𝑅𝑎 → 𝑎 = 𝑋) → (𝑌𝑅𝐴 → 𝐴 = 𝑋))
15	2, 14	syl 17	. 2 ⊢ (∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋) → (𝑌𝑅𝐴 → 𝐴 = 𝑋))
16		frege116.x	. . 3 ⊢ 𝑋 ∈ 𝑈
17		frege118.y	. . 3 ⊢ 𝑌 ∈ 𝑉
18	16, 17	frege119 42718	. 2 ⊢ ((∀𝑎(𝑌𝑅𝑎 → 𝑎 = 𝑋) → (𝑌𝑅𝐴 → 𝐴 = 𝑋)) → (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴 → 𝐴 = 𝑋))))
19	15, 18	ax-mp 5	1 ⊢ (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴 → 𝐴 = 𝑋)))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1539   = wceq 1541   ∈ wcel 2106  [wsbc 3776  ⦋csb 3892   class class class wbr 5147  ^◡ccnv 5674  Fun wfun 6534

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-frege1 42526  ax-frege2 42527  ax-frege8 42545  ax-frege52a 42593  ax-frege58b 42637

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ifp 1062  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-fun 6542

This theorem is referenced by:  frege121  42720